Recent zbMATH articles in MSC 65Mhttps://zbmath.org/atom/cc/65M2023-02-24T16:48:17.026759ZUnknown authorWerkzeugHeat kernels of generalized degenerate Schrödinger operators and Hardy spaceshttps://zbmath.org/1502.350502023-02-24T16:48:17.026759Z"The Anh Bui"https://zbmath.org/authors/?q=ai:the-anh-bui."Tan Duc Do"https://zbmath.org/authors/?q=ai:tan-duc-do."Nguyen Ngoc Trong"https://zbmath.org/authors/?q=ai:nguyen-ngoc-trong.Summary: Let \(\mathrm{L} = - \frac{ 1}{ {w}} \operatorname{div}({A}{\nabla} \mathrm{u}) + \mu\) be the generalized degenerate Schrödinger operator in \(\mathrm{L}_w^2( \mathbb{R}^d)\) with \(d \geq 3\) with suitable weight \(w\) and measure \(\mu \). The main aim of this paper is threefold. Firstly, we obtain an upper bound for the fundamental solution of the operator \(\mathrm{L}\). Secondly, we prove some estimates for the heat kernel of \(\mathrm{L}\) including an upper bound, the Hölder continuity, and a comparison estimate. Finally, we apply the results to study the maximal function characterization for the Hardy spaces associated to the critical function generated by the operator \(\mathrm{L}\).Methods for solving the problem of zonal electrophoresis with periodic initial datahttps://zbmath.org/1502.351092023-02-24T16:48:17.026759Z"Dolgikh, T. F."https://zbmath.org/authors/?q=ai:dolgikh.tatyana-fedorovnaSummary: We consider the problem of zonal electrophoresis of a two-component mixture with spatially periodic initial distribution of the mixture components. Two methods of solution are proposed: analytical (hodograph method) and numerical (method of finite volumes). A comparative analysis of the results obtained is performed.Recent advances and complex applications of the compressible ghost-fluid methodhttps://zbmath.org/1502.351152023-02-24T16:48:17.026759Z"Jöns, Steven"https://zbmath.org/authors/?q=ai:jons.steven"Müller, Christoph"https://zbmath.org/authors/?q=ai:muller.christoph-r"Zeifang, Jonas"https://zbmath.org/authors/?q=ai:zeifang.jonas"Munz, Claus-Dieter"https://zbmath.org/authors/?q=ai:munz.claus-dieterSummary: In this paper, improvements to a level-set ghost-fluid scheme in a high order discontinuous Galerkin framework with finite-volume sub-cells are presented. We propose the use of a path-conservative scheme for the level-set transport in both the discontinuous Galerkin and the finite-volume framework. Additionally, improvements regarding the curvature calculation and velocity extrapolation are described. The modified scheme is validated by a comparison of shock-drop and drop-drop interaction simulations from literature.
For the entire collection see [Zbl 1470.65004].Fourier analysis of the local discontinuous Galerkin method for the linearized KdV equationhttps://zbmath.org/1502.351452023-02-24T16:48:17.026759Z"Le Roux, Daniel Y."https://zbmath.org/authors/?q=ai:le-roux.daniel-ySummary: A Fourier/stability analysis of the third-order Korteweg-de Vries equation is presented subject to a class of local discontinuous Galerkin discretization using high-degree Lagrange polynomials. The selection of stability parameters involved in the method is made on the basis of the study of the higher frequency eigenmodes and the Fourier analysis. Explicit analytical dispersion relation and group velocity are obtained and the stability study of the discrete frequency is performed. The emergence of gaps in the imaginary part of the computed frequency is observed and studied for the first time to our knowledge. Further, a superconvergent result is demonstrated for the discrete frequency by obtaining an explicit analytical asymptotic formula for the latter.Analysis of a dimension splitting scheme for Maxwell equations with low regularity in heterogeneous mediahttps://zbmath.org/1502.351612023-02-24T16:48:17.026759Z"Zerulla, Konstantin"https://zbmath.org/authors/?q=ai:zerulla.konstantinSummary: We analyze a dimension splitting scheme for the time integration of linear Maxwell equations in a heterogeneous cuboid. The domain contains several homogeneous subcuboids and serves as a model for a rectangular embedded waveguide. Due to discontinuities of the material parameters and irregular initial data, the solution of the Maxwell system has regularity below \(H^1\). The splitting scheme is adapted to the arising singularities and is shown to converge with order one in \(L^2\). The error result only imposes assumptions on the model parameters and the initial data, but not on the unknown solution. To achieve this result, the regularity of the Maxwell system is analyzed in detail, giving rise to sharp explicit regularity statements. In particular, the regularity parameters are given in explicit terms of the largest jump of the material parameters. The analysis is based on semigroup theory, interpolation theory, and regularity analysis for elliptic transmission problems.Pseudospectral methods and iterative solvers for optimization problems from multiscale particle dynamicshttps://zbmath.org/1502.351692023-02-24T16:48:17.026759Z"Aduamoah, Mildred"https://zbmath.org/authors/?q=ai:aduamoah.mildred"Goddard, Benjamin D."https://zbmath.org/authors/?q=ai:goddard.benjamin-d"Pearson, John W."https://zbmath.org/authors/?q=ai:pearson.john-w"Roden, Jonna C."https://zbmath.org/authors/?q=ai:roden.jonna-cSummary: We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we investigate problems where the control acts as an advection `flow' vector or a source term of the partial differential equation, and the constraint is equipped with boundary conditions of Dirichlet or no-flux type. After deriving continuous first-order optimality conditions for such problems, we solve the resulting systems by developing a link with computational methods for statistical mechanics, deriving pseudospectral methods in space and time variables, and utilizing variants of existing fixed-point methods as well as a recently developed Newton-Krylov scheme. Numerical experiments indicate the effectiveness of our approach for a range of problem set-ups, boundary conditions, as well as regularization and model parameters, in both two and three dimensions. A key contribution is the provision of software which allows the discretization and solution of a range of optimization problems constrained by differential equations describing particle dynamics.Stratified radiative transfer in a fluid and numerical applications to Earth sciencehttps://zbmath.org/1502.351732023-02-24T16:48:17.026759Z"Golse, François"https://zbmath.org/authors/?q=ai:golse.francois"Pironneau, Olivier R."https://zbmath.org/authors/?q=ai:pironneau.olivier-rThe authors analyze radiative transfer equations alone or coupled with the temperature equation of a fluid. Assuming that a local thermodynamic equilibrium occurs at each point in the fluid, they write a kinetic equation for the radiative intensity \(I_{\nu}(x,\omega, t)\) at time \(t\) and position \(x\), in the direction \(\omega\), and for photons of frequency \(\nu\), in terms of the temperature field \(T(x,t)\) according to:
\[
\frac{1}{c}\partial_{t}I_{\nu}+\omega \cdot \nabla I_{\nu}+\rho =\overline{\kappa}_{\nu}a_{\nu}[I_{\nu}-\frac{1}{4\pi}\int_{\mathbb{S}^{2}}p_{\nu}(\omega, \omega^{\prime})I_{\nu}(\omega^{\prime})d\omega^{\prime}]=\rho \overline{\kappa}_{\nu}(1-a_{\nu})[B_{\nu}(T)I_{\nu}],
\]
where\ \(B_{\nu}(T)=\frac{2h\nu^{3}}{c^{2}[\exp (h\nu /kT)-1]}\) is the Planck function at temperature \(T\), \(h\) being the Planck constant, \(c\) the speed of light in the medium, and \(k\) the Boltzmann constant. This kinetic equation may be coupled with the fluid equations which are, in the case of an incompressible fluid, the Navier-Stokes equations:
\(\partial_{t}\rho +u\cdot \nabla \rho =0\), \(\nabla \cdot u=0\), \(\partial_{t}u+u\cdot \nabla u-\frac{\mu F}{\rho} \Delta u+\frac{1}{\rho}\nabla p=g\), where \(u\) is the velocity field, \(\rho\) the density, \(p\) the pressure, \(g\) the gravity, and \(\mu F\) the fluid viscosity. Combing these two equations and introducing some simplification, the authors obtain a system which is supplemented with appropriate initial and boundary conditions:
\(I_{\nu}(x,\omega, t)=Q_{\nu}(x,\omega, t)\), \(x\in \partial \Omega\), \(\omega \cdot n_{x}<0\), \(\nu >0\), \(u\mid_{\partial \Omega}=0=\frac{\partial T}{\partial n}\mid_{\partial \Omega}\), assuming, for instance, that the spatial domain is an open subset \(\Omega\) of \(\mathbb{R}^{3}\) with \(C^{1}\), or piecewise \(C^{1}\), boundary \(\partial \Omega\), \(n\) being the outward unit normal on \(\partial \Omega\).
They first consider the case where \(\kappa_{T}=0\), assuming that the fluid is at rest and an isotropic scattering: \(p_{\nu}(\mu, \mu^{\prime})=1\) at all frequencies \(\nu\). The general system becomes:
\((\mu \partial_{\tau}+\kappa_{\nu})I_{\nu}(\tau, \mu)=\kappa_{\nu}a_{\nu}J_{\nu}(\tau)+\kappa_{\nu}(1-a_{\nu})B_{\nu}(T(\tau)),\)
\(I_{\nu}(0,\mu)=Q_{\nu}^{+}(\mu)\), \(I_{\nu}(Z,-\mu)=Q_{\nu}^{-}(\mu)\), \(0<\mu <1\),
\(\int_{0}^{\infty}\kappa_{\nu}(1-a_{\nu})B_{\nu}(T(\tau))d\nu =\int_{0}^{\infty}\kappa_{\nu}(1-a_{\nu})J_{\nu}(\tau)d\nu,\)
with
\(Q_{\nu}^{-}(\mu)=Q^{-}(-\mu)B_{\nu}(T_{S})\) and \(J_{\nu}(\tau)=\frac{1}{2}\int_{-1}^{1}I_{\nu}(\tau, \mu)d\mu\).
The main purpose of the paper is here to propose a numerical resolution of this problem written in an equivalent way as
\[
(\mu \partial_{\tau}+\kappa_{\nu})I_{\nu}(\tau, \mu)=\kappa_{\nu}S_{\nu}[J]:=\kappa_{\nu}(a_{\nu}J_{\nu}(\tau)+\kappa_{\nu}(1-a_{\nu})B_{\nu}(T([J](\tau))),
\]
with the boundary conditions \(I_{\nu}(0,\mu)=Q_{\nu}^{+}(\mu)\), \(I_{\nu}(Z,-\mu)=Q_{\nu}^{-}(\mu)\), \(0<\mu <1\).
The authors indeed build the scheme
\((\mu \partial_{\tau}+\kappa_{\nu})I_{\nu}^{n+1}(\tau, \mu)=\kappa_{\nu}S_{\nu}[J^{n}]\),
\(I_{\nu}^{n+1}(0,\mu)=Q_{\nu}^{+}(\mu)\),
\(I_{\nu}^{n+1}(Z,-\mu)=Q_{\nu}^{-}(\mu)\), \(0<\mu <1\), and the main result proves for this system that if \(0<\kappa_{\nu}\leq \kappa_{M}\), \(0\leq a_{\nu}<1\) for all \(\nu >0\), \(Q_{\nu}^{\pm}(\mu)\) satisfy \(\mathcal{Q}=\frac{1}{2}\int_{0}^{\infty}\kappa_{\nu}\int_{0}^{1}(Q_{\nu}^{+}(\mu)+Q_{\nu}^{-}(\mu))\mu d\mu <\infty\), choosing \(I_{\nu}^{0}=0\) and \(T^{0}=0\), then \(I_{\nu}^{n}(\tau, \mu)\rightarrow I_{\nu}(\tau, \mu)\) and \(T^{n}(\tau)(=T[J_{\nu}^{n}])\rightarrow T(\tau)\), for \((\tau, \mu, \nu)\in (0,Z)\times (-1,1)\times (0,+\infty)\) as \(n\rightarrow \infty\), where \((I_{\nu},T)\) is a solution to the above problem.
This scheme is further proved to converge exponentially fast. For the proof, the authors use monotonicity properties of the sequences \(J_{\nu}^{n}\) and \(T^{n}\) and they rewrite the above system in an equivalent way. They also prove a maximum principle for this system. They then consider the problem obtained when replacing the isotropic scattering kernel by the Rayleigh phase function written in the case of slab symmetry as \(p(\mu, \mu^{\prime})=\frac{3}{16}(3-\mu^{2})+ \frac{3}{16}(3\mu^{2}-1)\mu^{\prime 2}\). They rewrite the problem and assuming almost the previously introduced hypotheses they prove a pointwise convergence result for the numerical scheme. A third model is finally introduced, the authors considering the case of a lake. They neglect the wind above the lake and assume that the sunlight hits the surface of the lake with a given energy. The depth of the lake is supposed to be uniform, thus considering the 3D domain: \(\Omega =O\times (0,Z)\) for some open set \(O\subset \mathbb{R}^{2}\) with \(C^{1}\) boundary, or piecewise \(C^{1}\), boundary. Assuming isotropic scattering with \(\rho\) constant and a steady fluid flow, they consider the system
\(\mu \partial_{z}I_{\nu}+\kappa_{\nu}=\kappa_{\nu}(1-a_{\nu})B_{\nu}(T)+\kappa_{\nu}a_{\nu}J_{\nu}(\tau)\),
\(u\cdot \nabla T-\frac{c_{P}}{c_{V}}\kappa_{T}\Delta T=\frac{4\pi}{\rho c_{V}}\int_{0}^{\infty}\kappa_{\nu}(1-a_{\nu})(J_{\nu}-B_{\nu}(T))d\nu\),
\(I_{\nu}\mid_{z=Z,\mu <0}=Q_{\nu}^{-}(x,y-\mu)\), \(I_{\nu}\mid_{z=Z,\mu >0}=Q_{\nu}^{+}(x,y,\mu)\), \(\frac{\partial T}{\partial n}\mid_{\partial \Omega}=0\).
Imposing appropriate hypotheses on the data, the authors prove a convergence result for the sequence of approximate solutions. The last section of the paper presents numerical results for the different models. The authors used either C++ or FreeFem languages for the numerical resolution and they draw some perspectives in the context of climate change.
Reviewer: Alain Brillard (Riedisheim)A tumor growth model with autophagy: the reaction-(cross-)diffusion system and its free boundary limithttps://zbmath.org/1502.351782023-02-24T16:48:17.026759Z"Dou, Xu'an"https://zbmath.org/authors/?q=ai:dou.xuan"Liu, Jian-Guo"https://zbmath.org/authors/?q=ai:liu.jian-guo"Zhou, Zhennan"https://zbmath.org/authors/?q=ai:zhou.zhennanSummary: In this paper, we propose a tumor growth model to incorporate and investigate the spatial effects of autophagy. The cells are classified into two phases: normal cells and autophagic cells, whose dynamics are also coupled with the nutrients. First, we construct a reaction-(cross-)diffusion system describing the evolution of cell densities, where the drift is determined by the negative gradient of the joint pressure, and the reaction terms manifest the unique mechanism of autophagy. Next, in the incompressible limit, such a cell density model naturally connects to a free boundary system, describing the geometric motion of the tumor region. Analyzing the free boundary model in a special case, we show that the ratio of the two phases of cells exponentially converges to a ``well-mixed'' limit. Within this ``well-mixed'' limit, we obtain an analytical solution of the free boundary system which indicates the exponential growth of the tumor size in the presence of autophagy in contrast to the linear growth without it. Numerical simulations are also provided to illustrate the analytical properties and to explore more scenarios.Density dependent diffusion models for the interaction of particle ensembles with boundarieshttps://zbmath.org/1502.351812023-02-24T16:48:17.026759Z"Weissen, Jennifer"https://zbmath.org/authors/?q=ai:weissen.jennifer"Göttlich, Simone"https://zbmath.org/authors/?q=ai:gottlich.simone"Armbruster, Dieter"https://zbmath.org/authors/?q=ai:armbruster.dieterSummary: The transition from a microscopic model for the movement of many particles to a macroscopic continuum model for a density flow is studied. The microscopic model for the free flow is completely deterministic, described by an interaction potential that leads to a coherent motion where all particles move in the same direction with the same speed known as a flock. Interaction of the flock with boundaries, obstacles and other flocks leads to a temporary destruction of the coherent motion that macroscopically can be modeled through density dependent diffusion. The resulting macroscopic model is an advection-diffusion equation for the particle density whose diffusion coefficient is density dependent. Examples describing i) the interaction of material flow on a conveyor belt with an obstacle that redirects or restricts the material flow and ii) the interaction of flocks (of fish or birds) with boundaries and iii) the scattering of two flocks as they bounce off each other are discussed. In each case, the advection-diffusion equation is strictly hyperbolic before and after the interaction while the interaction phase is described by a parabolic equation. A numerical algorithm to solve the advection-diffusion equation through the transition is presented.Solving nonlinear fractional partial differential equations using the Elzaki transform method and the homotopy perturbation methodhttps://zbmath.org/1502.351952023-02-24T16:48:17.026759Z"Mohamed, Mohamed. Z."https://zbmath.org/authors/?q=ai:mohamed.mohamed-z"Yousif, Mohammed"https://zbmath.org/authors/?q=ai:yousif.mohammed"Hamza, Amjad E."https://zbmath.org/authors/?q=ai:hamza.amjad-e(no abstract)The quasi-reversibility method for an inverse source problem for time-space fractional parabolic equationshttps://zbmath.org/1502.352032023-02-24T16:48:17.026759Z"Duc, Nguyen Van"https://zbmath.org/authors/?q=ai:duc.nguyen-van"Thang, Nguyen Van"https://zbmath.org/authors/?q=ai:thang.nguyen-van"Thành, Nguyen Trung"https://zbmath.org/authors/?q=ai:thanh.nguyen-trungSummary: In this paper, we apply the quasi-reversibility method to solve an inverse source problem for a time-space fractional parabolic equation. Hölder-type error estimates for the regularized solutions are proved for both a priori and a posteriori regularization parameter choice rules. The theoretical error estimates are confirmed with numerical tests for one and two dimensional equations.Stochastic processes with applications in physics and insurance. (Abstract of thesis)https://zbmath.org/1502.600652023-02-24T16:48:17.026759Z"He, Yue"https://zbmath.org/authors/?q=ai:he.yueFrom the text: Having been settled around the 1950s and matured afterwards, the theory of stochastic processes has been a common tool for describing
time-evolving randomness in various systems, with applications in a number of disciplines, including physics, biology and finance.
This thesis consists of five chapters, with Chapters 1, 2 and 3 investigating stochastic processes with applications in physics, and Chapters 4 and 5 in insurance mathematics.On well-posedness of stochastic anisotropic \(p\)-Laplace equation driven by Lévy noisehttps://zbmath.org/1502.601042023-02-24T16:48:17.026759Z"Neelima"https://zbmath.org/authors/?q=ai:neelima.c-a|neelima.dSummary: \textit{J. L. Lions} [Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod; Paris: Gauthier-Villars (1969; Zbl 0189.40603)] solved the anisotropic \(p\)-Laplace equation in deterministic setting by considering the anisotropic \(p\)-Laplace operator in \(d\)-dimensions as a sum of \(d\) monotone coercive operators each defined on a different space. Motivated by this example, we prove existence and uniqueness results for a large class of stochastic partial differential equations(SPDEs) driven by Lévy noise when the operator appearing in the bounded variation term is a sum of operators having different analytic and growth properties. Further, the operators are allowed to be locally monotone without explicitly restricting the growth of the operators appearing in the stochastic integrals. This has been done by identifying an appropriate coercivity condition. As a consequence, well-posedness of Lévy driven stochastic Anisotropic \(p\)-Laplace equation has been shown. Our framework is most general till date. Many popular SPDEs appearing in real world models such as the stochastic Ginzburg-Landau equation and stochastic Swift-Hohenberg equation, both driven by Lévy noise, fit in our setting. These equations are not covered by the corresponding results in the literature.Finite element analysis. With numeric and symbolic Matlabhttps://zbmath.org/1502.650012023-02-24T16:48:17.026759Z"Akin, John E."https://zbmath.org/authors/?q=ai:akin.john-edwardPublisher's description: This comprehensive compendium presents the detailed theory, implementation and application of finite element analysis via heavily commented Matlab scripts. The book includes over 110 examples of the methods, and has a very detailed subject index. It uniquely illustrates the use of symbolic Matlab capabilities to derive element interpolation functions and to analytically integrated complicated element matrices.
The useful volume text is suitable as a reference on finite element methods and efficient Matlab programming. Chapters prominently end with a detailed summary of the important features and tables of useful finite element matrices. It can be used as the textbook for introductory, intermediate, or advanced courses utilizing numerically integrated and curvilinear element.
Contents:
\begin{itemize}
\item Overview
\item Calculus review
\item Differential equations terminology
\item Parametric interpolation
\item Numerical integration
\item Equivalent integral forms
\item Matrix procedures
\item Truss analysis
\item Applications of 1-D Hermite elements
\item Frame analysis
\item Elasticity
\item Eigen-analysis
\item Transient and dynamic solutions
\end{itemize}Boundary elements and other mesh reduction methods XLV. Selected papers based on the presentations at the 45th international conference (BEM/MRM), Tallinn, Estonia, May 24--26, 2022https://zbmath.org/1502.650022023-02-24T16:48:17.026759ZPublisher's description: Advances in techniques that reduce or eliminate the type of meshes associated with finite elements or finite differences are reported in the papers that form this volume.
As design, analysis and manufacture become more integrated, the chances are that software users will be less aware of the capabilities of the analytical techniques that are at the core of the process. This reinforces the need to retain expertise in certain specialised areas of numerical methods, such as BEM/MRM, to ensure that all new tools perform satisfactorily within the aforementioned integrated process.
The maturity of BEM since 1978 has resulted in a substantial number of industrial applications of the method; this demonstrates its accuracy, robustness and ease of use. The range of applications still needs to be widened, taking into account the potentialities of the Mesh Reduction techniques in general.
The included papers originate from the 45th conference on Boundary Elements and other Mesh Reduction Methods (BEM/MRM) and describe theoretical developments and new formulations, helping to expand the range of applications as well as the type of modelled materials in response to the requirements of contemporary industrial and professional environments.
The articles of this volume will not be indexed individually. For the preceding conference see [Zbl 1469.65008].Maximum time step for high order BDF methods applied to gradient flowshttps://zbmath.org/1502.650472023-02-24T16:48:17.026759Z"Pierre, Morgan"https://zbmath.org/authors/?q=ai:pierre.morganSummary: For a backward differentiation formula (BDF) applied to the gradient flow of a semiconvex function, quadratic stability implies gradient stability. Namely, it is possible to build a Lyapunov functional for the discrete-in-time dynamical system, with a restriction on the time step. The maximum time step which can be derived from quadratic stability has previously been obtained for the BDF1, BDF2 and BDF3 schemes. Here, we compute it for the BDF4 and BDF5 methods. We also prove that the BDF6 scheme is not quadratically stable. Our results are based on the tools developed by Dahlquist and other authors to show the equivalence of \(A\)-stability and \(G\)-stability. We give several applications of gradient stability to the asymptotic behaviour of sequences generated by BDF schemes.Convergent semi-explicit scheme to a non-linear eikonal systemhttps://zbmath.org/1502.650532023-02-24T16:48:17.026759Z"Al Zohbi, Marya"https://zbmath.org/authors/?q=ai:al-zohbi.marya"El Hajj, Ahmad"https://zbmath.org/authors/?q=ai:el-hajj.ahmad"Jazar, Mustapha"https://zbmath.org/authors/?q=ai:jazar.mustaphaSummary: We consider a system of non-linear eikonal equations in one space dimension that describes the evolution of interfaces moving with non-signed strongly coupled velocities. We have recently proven the global existence and uniqueness of viscosity solutions for this system, under a \textit{BV} estimate. In this paper, we propose a semi-explicit scheme that satisfies the same \textit{BV} estimate proven in the continuous case, at the discrete level, and we show that a certain linear interpolation of the discrete solution to the scheme converges to a viscosity solution of the main system considered. We also provide some numerical simulations in the case of dislocation dynamics.Comparison of exponential integrators and traditional time integration schemes for the shallow water equationshttps://zbmath.org/1502.650542023-02-24T16:48:17.026759Z"Brachet, Matthieu"https://zbmath.org/authors/?q=ai:brachet.matthieu"Debreu, Laurent"https://zbmath.org/authors/?q=ai:debreu.laurent"Eldred, Christopher"https://zbmath.org/authors/?q=ai:eldred.christopherThe authors investigate several time integration schemes when applied to the shallow water equations. This set of equations is accurate enough for the modeling of a shallow ocean and is also relevant to study as it is the one solved for the barotropic (i.e. vertically averaged) component of a three-dimensional ocean model. They analyze different time stepping algorithms for the linearized shallow water equations. High-order explicit schemes are accurate but the time step is constrained by the Courant-Friedrichs-Lewy stability condition. Implicit schemes can be unconditionally stable but, in practice lack accuracy when used with large time steps. The authors propose a detailed comparison of such classical schemes with exponential integrators. The accuracy and the computational costs are analyzed in different configurations.
Reviewer: Qifeng Zhang (Hangzhou)A multi-splitting method to solve 2D parabolic reaction-diffusion singularly perturbed systemshttps://zbmath.org/1502.650552023-02-24T16:48:17.026759Z"Clavero, C."https://zbmath.org/authors/?q=ai:clavero.carmelo"Jorge, J. C."https://zbmath.org/authors/?q=ai:jorge.juan-carlosSummary: In this paper we design and analyze a numerical method to solve a type of reaction-diffusion 2D parabolic singularly perturbed systems. The method combines the central finite difference scheme on an appropriate piecewise uniform mesh of Shishkin-type to discretize in space, and the fractional implicit Euler method together with a splitting by directions and components of the reaction-diffusion operator to integrate in time. We prove that the method is uniformly convergent of first order in time and almost second order in space. The use of this time integration technique has the advantage that only tridiagonal linear systems must be solved to obtain the numerical solution at each time step; because of this, our method provides a remarkable reduction of computational cost, in comparison with other implicit methods which have been previously proposed for the same type of problems. Full details of the uniform convergence are given only for systems with two equations; nevertheless, our ideas can be easily extended to systems with an arbitrary number of equations as it is shown in the numerical experiences performed. The numerical results show in practice the qualities of our proposal.Sparse grid reconstructions for particle-in-cell methodshttps://zbmath.org/1502.650562023-02-24T16:48:17.026759Z"Deluzet, Fabrice"https://zbmath.org/authors/?q=ai:deluzet.fabrice"Fubiani, Gwenael"https://zbmath.org/authors/?q=ai:fubiani.gwenael"Garrigues, Laurent"https://zbmath.org/authors/?q=ai:garrigues.laurent"Guillet, Clément"https://zbmath.org/authors/?q=ai:guillet.clement"Narski, Jacek"https://zbmath.org/authors/?q=ai:narski.jacekSummary: In this article, we propose and analyse Particle-In-Cell (PIC) methods embedding sparse grid reconstructions such as those introduced in [\textit{L. F. Ricketson} and \textit{A. J. Cerfon}, ``Sparse grid techniques for particle-in-cell schemes'', Plasma Phys. Control. Fusion 59, No. 2, Article ID 024002, 19 p. (2016; \url{doi:10.1088/1361-6587/59/2/024002})] and [\textit{S. Muralikrishnan} et al., ``Sparse grid-based adaptive noise reduction strategy for particle-in-cell schemes'', J. Comput. Phys. X 11, Article ID 100094, 31 p. (2021; \url{doi:10.1016/j.jcpx.2021.100094})]. The sparse grid reconstructions offer a significant improvement on the statistical error of PIC schemes as well as a reduction in the complexity of the problem providing the electric field. Main results on the convergence of the electric field interpolant and conservation properties are provided in this paper. Besides, tailored sparse grid reconstructions, in the frame of the offset combination technique, are proposed to introduce PIC methods with improved efficiency. The methods are assessed numerically and compared to existing PIC schemes thanks to classical benchmarks with remarkable prospects for three dimensional computations.High-order structure-preserving Du Fort-Frankel schemes and their analyses for the nonlinear Schrödinger equation with wave operatorhttps://zbmath.org/1502.650572023-02-24T16:48:17.026759Z"Deng, Dingwen"https://zbmath.org/authors/?q=ai:deng.dingwen"Li, Zhijun"https://zbmath.org/authors/?q=ai:li.zhijun.1Summary: Du Fort-Frankel-type finite difference methods (DFFT-FDMs) are famous for good stability and easy implementation. In this study, by a perfect combination of the classical fourth-order difference to approximate the second-order spatial derivatives with the idea of DFFT-FDMs, a class of high-order structure-preserving DFFT-FDMs (SP-DFFT-FDMs) are firstly developed for solving the periodic initial-boundary value problems (PIBVPs) of one-dimensional (1D) and two-dimensional (2D) nonlinear Schrödinger equations with wave operator (NLSW), respectively. By using the discrete energy method, it is shown that their solutions satisfy the discrete energy- and mass-conservative laws, and are conditionally convergent with an order of \(\mathcal{O} (\tau^2 + h_x^4 + (\frac{\tau}{h_x} )^2)\) and \(\mathcal{O} (\tau^2 + h_x^4 + h_y^4 + (\frac{\tau}{h_x})^2 + (\frac{\tau}{h_y})^2)\) in the discrete \(H^1\)-norm, respectively. Here, \( \tau\) denotes time step size, while, \( h_x\) and \(h_y\) represent spatial mesh sizes in \(x\) and \(y\) directions, respectively. Then, by supplementing a stabilized term, a type of stabilized SP-DFFT-FDMs are devised. They not only preserve the discrete energy- and mass-conservation laws, but also own much better stability than original SP-DFFT-FDMs. Finally, numerical results confirm the theoretical findings, and the efficiency of our algorithms.The studies of the linearly modified energy-preserving finite difference methods applied to solve two-dimensional nonlinear coupled wave equationshttps://zbmath.org/1502.650582023-02-24T16:48:17.026759Z"Deng, Dingwen"https://zbmath.org/authors/?q=ai:deng.dingwen"Wu, Qiang"https://zbmath.org/authors/?q=ai:wu.qiang|wu.qiang.1Summary: In this paper, four linearly energy-preserving finite difference methods (EP-FDMs) are designed for two-dimensional (2D) nonlinear coupled sine-Gordon equations (CSGEs) and coupled Klein-Gordon equations (CKGEs) using the invariant energy quadratization method (IEQM). The 1st EP-FDM is designed by first introducing two auxiliary functions to rewrite the original problems into the new system only including the 1st-order temporal derivatives, and then applying Crank-Nicolson (C-N) method and 2nd-order centered difference methods for the discretizations of temporal and spatial derivatives, respectively. The 2nd EP-FDM is directly devised based on the uses of 2nd-order centered difference methods to approximate 2nd-order temporal and spatial derivatives. The 1st and 2nd EP-FDMs need numerical solutions of the algebraic system with variable coefficient matrices at each time level. By modifying the 2nd EP-FDM, the 3rd EP-FDM, which is implemented by computing the system of algebraic equations with constant coefficient matrices at each time level, is developed. Finally, an energy-preserving alternating direction implicit (ADI) finite difference method (EP-ADI-FDM) is established by a combination of ADI method with the 3rd EP-FDM. By using the discrete energy method, it is shown that they are all uniquely solvable, and their solutions have a convergent rate of \({\mathscr{O}}(\Delta t^2+{h^2_x}+{h^2_y})\) in \(H^1\)-norm and satisfy the discrete conservative laws. Numerical results show the efficiency and accuracy of them.Fast and reliable transient simulation and continuous optimization of large-scale gas networkshttps://zbmath.org/1502.650592023-02-24T16:48:17.026759Z"Domschke, Pia"https://zbmath.org/authors/?q=ai:domschke.pia"Kolb, Oliver"https://zbmath.org/authors/?q=ai:kolb.oliver"Lang, Jens"https://zbmath.org/authors/?q=ai:lang.jensSummary: We are concerned with the simulation and optimization of large-scale gas pipeline systems in an error-controlled environment. The gas flow dynamics is locally approximated by sufficiently accurate physical models taken from a hierarchy of decreasing complexity and varying over time. Feasible work regions of compressor stations consisting of several turbo compressors are included by semiconvex approximations of aggregated characteristic fields. A discrete adjoint approach within a first-discretize-then-optimize strategy is proposed and a sequential quadratic programming with an active set strategy is applied to solve the nonlinear constrained optimization problems resulting from a validation of nominations. The method proposed here accelerates the computation of near-term forecasts of sudden changes in the gas management and allows for an economic control of intra-day gas flow schedules in large networks. Case studies for real gas pipeline systems show the remarkable performance of the new method.Weak versus strong wall boundary conditions for the incompressible Navier-Stokes equationshttps://zbmath.org/1502.650602023-02-24T16:48:17.026759Z"Eriksson, Gustav"https://zbmath.org/authors/?q=ai:eriksson.gustav"Mattsson, Ken"https://zbmath.org/authors/?q=ai:mattsson.kenA high-order summation-by-parts (SBP) finite difference method is developed for the velocity-pressure formulation of the incompressible Navier-Stokes equations. The paper's primary goal is to compare the simultaneous approximation term (SAT) or the projection (P) method by imposing Dirichlet and no-slip wall boundary conditions. Both discretizations of the incompressible Navier-Stokes equations with Dirichlet boundary conditions are presented by enforcing the divergence-free condition and deriving consistent pressure boundary data. The resulting schemes are proven to be stable using the energy method. Furthermore, new optimal highly efficient second-derivative SBP operators are derived. The efficiency and accuracy of the methods are demonstrated on three benchmark problems: the Taylor-Green vortex flow, the lid-driven cavity, and the backwards-facing step. The accuracy and convergence properties of the schemes are verified for the traditional and novel optimal SBP operators.
Reviewer: Bülent Karasözen (Ankara)Inverses of SBP-SAT finite difference operators approximating the first and second derivativehttps://zbmath.org/1502.650612023-02-24T16:48:17.026759Z"Eriksson, Sofia"https://zbmath.org/authors/?q=ai:eriksson.sofiaSummary: The scalar, one-dimensional advection equation and heat equation are considered. These equations are discretized in space, using a finite difference method satisfying summation-by-parts (SBP) properties. To impose the boundary conditions, we use a penalty method called simultaneous approximation term (SAT). Together, this gives rise to two semi-discrete schemes where the discretization matrices approximate the first and the second derivative operators, respectively. The discretization matrices depend on free parameters from the SAT treatment. We derive the inverses of the discretization matrices, interpreting them as discrete Green's functions. In this direct way, we also find out precisely which choices of SAT parameters that make the discretization matrices singular. In the second derivative case, it is shown that if the penalty parameters are chosen such that the semi-discrete scheme is dual consistent, the discretization matrix can become singular even when the scheme is energy stable. The inverse formulas hold for SBP-SAT operators of arbitrary order of accuracy. For second and fourth order accurate operators, the inverses are provided explicitly.A monotone, second order accurate scheme for curvature motionhttps://zbmath.org/1502.650622023-02-24T16:48:17.026759Z"Esedoḡlu, Selim"https://zbmath.org/authors/?q=ai:esedoglu.selim"Guo, Jiajia"https://zbmath.org/authors/?q=ai:guo.jiajiaThis article discusses a second order accurate in time numerical scheme for curve shortening flow in the plane that is unconditionally monotone. It is a variant of threshold dynamics, a class of algorithms in the spirit of the level set method that represent interfaces implicitly. The main novelty of the current work is the monotonicity feature: it is possible to preserve the comparison principle of the exact evolution while achieving second order in time consistency. As a consequence of monotonicity, convergence to the viscosity solution of curve shortening is deduced. Numerical experiments are included to support the theoretical findings.
Reviewer: Marius Ghergu (Dublin)Energy plus maximum bound preserving Runge-Kutta methods for the Allen-Cahn equationhttps://zbmath.org/1502.650632023-02-24T16:48:17.026759Z"Fu, Zhaohui"https://zbmath.org/authors/?q=ai:fu.zhaohui"Tang, Tao"https://zbmath.org/authors/?q=ai:tang.tao"Yang, Jiang"https://zbmath.org/authors/?q=ai:yang.jiangSummary: It is difficult to design high order numerical schemes which could preserve both the maximum bound property (MBP) and energy dissipation law for certain phase field equations. Strong stability preserving (SSP) Runge-Kutta methods have been developed for numerical solution of \textit{hyperbolic partial differential equations} in the past few decades, where strong stability means the non-increasing of the maximum bound of the underlying solutions. However, existing framework of SSP RK methods can not handle nonlinear stabilities like energy dissipation law. The aim of this work is to extend this SSP theory to deal with the nonlinear phase field equation of the Allen-Cahn-type which typically satisfies both maximum bound preserving (MBP) and energy dissipation law. More precisely, for Runge-Kutta time discretizations, we first derive a general necessary and sufficient condition under which MBP is satisfied; and we further provide a necessary condition under which the MBP scheme satisfies energy dissipation.Characteristic methods for thermal convection problems with infinite Prandtl number on nonuniform staggered gridshttps://zbmath.org/1502.650642023-02-24T16:48:17.026759Z"Guo, Qing"https://zbmath.org/authors/?q=ai:guo.qing"Rui, Hongxing"https://zbmath.org/authors/?q=ai:rui.hongxingThe authors are concerned with some finite difference methods in order to approximate the solutions of some thermal convection models with infinite Prandtl number on non-uniform staggered grids. The model consists of partial differential equations for the velocity field, pressure and temperature. Unfortunately, there is no indication that the model is well posed. Backward Euler for time derivative along with a finite difference method based on characteristics is used in order to approximate the model's solutions. The authors try some error estimates and present a non conclusive numerical example.
Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)Block-centered local refinement methods for the time-fractional equationshttps://zbmath.org/1502.650652023-02-24T16:48:17.026759Z"Guo, Qing"https://zbmath.org/authors/?q=ai:guo.qing"Rui, Hongxing"https://zbmath.org/authors/?q=ai:rui.hongxingSummary: In this paper, we present and analyze two block-centered local refinement (BLR) methods for solving the time-fractional equations. The main difference between the two methods is that different approximation methods are used for the value of the pressure at the slave nodes. One adopts a piecewise constant interpolation approximation (PCIA) method, which is called simple block-centered local refinement (S-BLR) method. The other utilizes a piecewise linear interpolation approximation (PLIA) method, which called more accurate block-centered local refinement (MA-BLR) method. The stability analysis is proved carefully. It is estimated that the discrete \(L^2\) errors for the velocity and pressure are \(O(\Delta t^{2-\alpha}+h^{3/2})\) and \(O(\Delta t^{2-\alpha}+h^{3/2})\) in use of S-BLR and MA-BLR methods, respectively. Where \(\Delta t\) is the time step and \(h\) is the maximal mesh size. These error estimate results are all established on locally refinement composite grids. Finally, a numerical experiment is presented to show that the convergence rates are in agreement with the theoretical analysis.Positivity-preserving and unconditionally energy stable numerical schemes for MEMS modelhttps://zbmath.org/1502.650662023-02-24T16:48:17.026759Z"Hou, Dianming"https://zbmath.org/authors/?q=ai:hou.dianming"Wang, Hui"https://zbmath.org/authors/?q=ai:wang.hui.40"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.6|zhang.chao.2|zhang.chao.7|zhang.chao.3|zhang.chao.8|zhang.chao.5|zhang.chao.1|zhang.chaoSummary: In this paper, we propose and analyze the positivity-preserving, unconditionally energy stable, and linear second order fully discrete schemes for Micro-Electromechanical system (MEMS). More precisely, we use the first order backward difference formulation (BDF1) and second order Crank-Nicolson (CN) formulation for the temporal discretization, and the central finite difference method for spatial discretization. A variant of the exponential scalar auxiliary variable (ESAV) approach is involved in our numerical schemes to deal with the singular nonlinear term. The unconditional energy stability of the numerical schemes is rigorously proved, without any restriction for the time step sizes. Furthermore, we derive that the numerical solutions always preserve the positivity property of the MEMS model, that is the distance variable is always between 0 and the steady solution, at a point-wise level. A series of numerical simulations are presented to demonstrate the positivity and energy stability of our numerical schemes.High-accurate and robust conservative remapping combining polynomial and hyperbolic tangent reconstructionshttps://zbmath.org/1502.650672023-02-24T16:48:17.026759Z"Kucharik, Milan"https://zbmath.org/authors/?q=ai:kucharik.milan"Loubère, Raphaël"https://zbmath.org/authors/?q=ai:loubere.raphaelSummary: In this article we present a 1D single-material conservative remapping method that relies on high accurate reconstructions: polynomial (\(\mathbb{P}_4,\mathbb{P}_1\) with slope limiter) and non-linear hyperbolic tangent (THINC) representations. Such remapping procedure is intended to be used pairwise with a cell-centered Lagrangian scheme along with a rezone strategy to build a so-called indirect Arbitrary-Lagrangian-Eulerian scheme. Most of practically used Lagrangian schemes are second-order accurate. The goal of this work is to handle with accuracy contact using THINC reconstructions. At the same time, the smooth part of the solution is dealt with quartic polynomials, resulting locally in fifth order accurate remapping method. To ensure robustness, TVD-like reconstructions (\(\mathbb{P}_1\) with slope limiter) are employed otherwise. A simple feature tracking algorithm is designed to assign a reconstruction type per cell (\(\mathbb{P}_4, \mathbb{P}_1^{\lim}\) or THINC). This tracking algorithm is based on the nature of the contact waves which are traveling at the fluid velocity, while the shocks are compressive and detectable by following a change of cell volumes. Numerical results assess the behavior of such a remapping method on pure remapping problems of a scalar quantity and in the context of the full hydrodynamics equations. The associated indirect cell-centered ALE numerical scheme is run and produces numerical results that are presented to assess the extreme accuracy gained by such a remapping procedure employing a mix of reconstruction types.Effective time step analysis of convex splitting schemes for the Swift-Hohenberg equationhttps://zbmath.org/1502.650682023-02-24T16:48:17.026759Z"Lee, Seunggyu"https://zbmath.org/authors/?q=ai:lee.seunggyu-lee"Yoon, Sungha"https://zbmath.org/authors/?q=ai:yoon.sungha"Kim, Junseok"https://zbmath.org/authors/?q=ai:kim.junseokSummary: We study the effective temporal step size of convex splitting schemes for the Swift-Hohenberg (SH) equation, which models the pattern formation in various physical systems. The convex splitting scheme is one of the most well-known numerical approaches with an unconditional stability for solving a gradient flow. Its stability, solvability, and convergence have been actively studied; however, only a few studies have analyzed the time step re-scaling phenomenon for certain applications. In this paper, we present effective time step formulations for different convex splitting methods. Several numerical simulations are conducted to confirm the effective time step analysis.A fast stable accurate artificial boundary condition for the linearized Green-Naghdi systemhttps://zbmath.org/1502.650692023-02-24T16:48:17.026759Z"Pang, Gang"https://zbmath.org/authors/?q=ai:pang.gang"Ji, Songsong"https://zbmath.org/authors/?q=ai:ji.songsongSummary: In this paper, we use a fully discrete Crank-Nicolson scheme to solve the Cauchy problem for one-dimensional linearized Green-Naghdi system with fast convolution boundary condition which is derived through the Padé approximation for the square root function. We also introduce a constant damping term to guarantee the stability. While the damping term meets certain conditions, the stability is proved for the numerical solutions with the fast convolution boundary condition. Therefore, the difficulty of numerical instability which rises in [\textit{M. Kazakova} and \textit{P. Noble}, SIAM J. Numer. Anal. 58, No. 1, 657--683 (2020; Zbl 1434.65121)] is overcome. The computational cost of the convolution integral is reduced from \(\mathcal{O}(N^2)\) to \(\mathcal{O}(N\ln (N))\) for the total number of time steps \(N\). Numerical examples verify the theoretical results and demonstrate the performance for the fast numerical method.Error estimates for the scalar auxiliary variable (SAV) schemes to the modified phase field crystal equationhttps://zbmath.org/1502.650702023-02-24T16:48:17.026759Z"Qi, Longzhao"https://zbmath.org/authors/?q=ai:qi.longzhao"Hou, Yanren"https://zbmath.org/authors/?q=ai:hou.yanrenSummary: We design first-order and second-order time-stepping schemes for the modified phase field crystal model based on the scalar auxiliary variable method in this work. The model is a nonlinear sixth-order damped wave equation that includes both elastic interactions and diffusive dynamics. Our schemes are linear and satisfy the unconditional energy stability with respect to pseudo energy. We also rigorously estimate the errors of the numerical schemes. Finally, some numerical tests are presented to validate our theoretical results.A novel high-order numerical scheme and its analysis for the two-dimensional time-fractional reaction-subdiffusion equationhttps://zbmath.org/1502.650712023-02-24T16:48:17.026759Z"Roul, Pradip"https://zbmath.org/authors/?q=ai:roul.pradip"Rohil, Vikas"https://zbmath.org/authors/?q=ai:rohil.vikasSummary: This paper deals with the design and analysis of a high-order numerical scheme for the two-dimensional time-fractional reaction-subdiffusion equation. The time-fractional derivative of order \(\alpha\), \((\alpha \in (0,1))\) in the model problem is approximated by means of the \(L2\)-\(1_\sigma\) scheme and the space derivatives are discretized by means of a compact alternating direction implicit (ADI) finite difference scheme. Convergence and stability of the method are analyzed. Five numerical examples are provided to demonstrate the applicability and accuracy of the method. It is shown that the method is unconditionally stable and is of \(\mathcal{O}({\Delta} t^{1+\alpha}+{h_x^4}+{h_y^4})\) accuracy, where \(\Delta t\) is the temporal step size and \(h_x\) and \(h_y\) are the spatial step sizes. The computed results are in good agreement with the theoretical analysis. Moreover, the comparison with the corresponding results of existing methods demonstrates that our method has advantages in convergence order and error improvement when solving the time-fractional reaction-subdiffusion equation. The CPU time consumed by the proposed method is provided.Accurate numerical scheme for solving fractional diffusion-wave two-step model for nanoscale heat conductionhttps://zbmath.org/1502.650722023-02-24T16:48:17.026759Z"Shen, Shujun"https://zbmath.org/authors/?q=ai:shen.shujun"Dai, Weizhong"https://zbmath.org/authors/?q=ai:dai.weizhong"Liu, Qingxia"https://zbmath.org/authors/?q=ai:liu.qingxia"Zhuang, Pinghui"https://zbmath.org/authors/?q=ai:zhuang.pinghuiSummary: Recently, we have presented a fractional two-step model and its numerical method for nanoscale heat conduction. The model was obtained by introducing the Knudsen number \((K_n)\) and the fractional-order \(( 0 < \alpha < 1)\) derivative in time to the parabolic two-step energy transport equations. For the case of \(0 < \alpha < 1\), the model governs the ultraslow diffusion and we may call it the sub-diffusion two-step model. In this article, we extend this study to the case of \(1 < \alpha < 2\) for that we may call the diffusion-wave two-step model, which can govern the intermediate processes for nanoscale heat conduction. The numerical scheme is developed based on the \(L_1\) approximation for fractional derivatives and the compact finite difference scheme for spatial derivative. Stability and convergence of the obtained numerical scheme are analyzed theoretically. We finally test the accuracy and applicability of the new model and its numerical scheme by three examples. By changing values of the Knudsen number and fractional-order derivative as well as the parameter in the boundary condition, the simulation could be a tool for analyzing the nanoscale heat conduction in intermediate processes such as porous and impure thin films exposed to ultrashort-pulsed lasers.Implicit-explicit-compact methods for advection diffusion reaction equationshttps://zbmath.org/1502.650732023-02-24T16:48:17.026759Z"Singh, S."https://zbmath.org/authors/?q=ai:singh.santokh|singh.shyam|singh.sanjay-kumar.2|singh.sandeep-kumar.2|singh.sakshi|singh.sushil|singh.saurabh|singh.sujeet-kumar.1|singh.sudhir|singh.swadesh|singh.shailendra|singh.surya-prakash.1|singh.satya-prakash.3|singh.sudarshan-pal.1|singh.sanjeev.1|singh.satyajeet|singh.satender|singh.shreya|singh.sargun|singh.sinam-ajitkumar|singh.sooraj|singh.shiva-n|singh.s-robertson|singh.sompal|singh.shantanu|singh.sagar|singh.suryadev-pratap|singh.suram|singh.shveta|singh.suneet|singh.satinder-pal|singh.satnesh|singh.surjan|singh.satnam|singh.satyabir|singh.shashank-sheshar|singh.sampooran|singh.sangita|singh.sanjeet|singh.satyvir|singh.shreekant|singh.surbjeet|singh.sukhbir|singh.smt-lalita|singh.sanjiv|singh.samayveer|singh.shaligram|singh.sunil|singh.shivangi|singh.somya|singh.sangeeta|singh.sube|singh.shubham|singh.sukhdeep|singh.sudhanshu-v|singh.sonza|singh.shubha|singh.satpal|singh.sungeeta|singh.sudhakar|singh.sukhdip|singh.sumeeta|singh.sumit.1|singh.siddharth|singh.sb|singh.sukhvir|singh.soniya|singh.swapnil|singh.suruchi|singh.sonalika|singh.shalini|singh.sukhdev|singh.s-n|singh.soumyendra|singh.soibam-shyamchand|singh.shekhawat-ashok|singh.samar|singh.shripal|singh.sanasam-ranbir|singh.shubh-narayan|singh.shishpal|singh.somorjit-konthoujam|singh.sudhansu-sekhar|singh.subedar|singh.shailza|singh.sumita|singh.sunder|sonia.|singh.sumeetpal-s|singh.sameer|singh.sandip|singh.surya-narayan|singh.s-surendra|singh.sita-ram|singh.satvik|singh.sunita|singh.shonal|singh.sehijpal|singh.siddhartha|singh.s-subhaschandra|singh.shivani|singh.swarn|singh.sukhminder|singh.sanjay-r|singh.sudha|singh.sarjit|singh.s-r-j|singh.sukhdave|singh.saroj|singh.sukhwinder|singh.shilpi|singh.shivdeep|singh.sukhhminder|kainth.surinder-pal-singh|singh.shobh-nath|singh.shikha|singh.s-surachandra|singh.saumya|singh.saurav|singh.suneel|singh.satbir|singh.sonam|singh.surender|singh.s-somorendro|singh.sawai|singh.sripal|singh.shashi-shekhar-kr|singh.salam-shantikumar|singh.surya-pratap|singh.shanker|singh.surindar|singh.sahjendra-n|singh.surya-p-n|singh.seema|singh.satvir|singh.shashank-vikram|singh.shri-dhar|singh.suresh-b|singh.subhash-ranjan|singh.sukh-mahendra|singh.santosh-kumar|singh.surendra-pratap|singh.shiv-raj|singh.surinder|singh.shweta|singh.sukhveer|singh.sadhna|singh.sumeet|singh.sartajvir|singh.saraswati-p|singh.sonika|singh.suraj-bhan|singh.soibam-nabachandra|singh.sanchit|singh.supreet|singh.sarita|singh.surjeet|singh.shiwani|singh.shalabh|singh.satyanand|singh.s-i|singh.sayuri|singh.shesh-nath|singh.sudhanshu-s|singh.shambhavi|singh.som-prakash|singh.subhash-chandra|singh.sarva-jit|singh.simmi|singh.sarbjeet|singh.saket-bihari|singh.suprit|singh.sukjith|singh.sadhu|singh.satwinder-jit|singh.shri-krishna|singh.satya-veer|singh.sanasam-sarat|singh.sadanand|singh.shiva-raj|singh.shwetank|singh.saban|singh.sundeep|singh.sidhnath|singh.simran-j|singh.soni|singh.singh-r|singh.sarjinder|singh.sukhjit|singh.surjit|singh.sulinder|singh.shailendra-narayan|singh.sachchidanand|singh.suprava|singh.shipra"Bansal, D."https://zbmath.org/authors/?q=ai:bansal.dinesh-rani|bansal.deepak|bansal.d-k|bansal.dipanshu"Kaur, G."https://zbmath.org/authors/?q=ai:kaur.gagandeep|kaur.gurwinder|kaur.gurleen|kaur.gurpreet|kaur.gurwinderpal|kaur.gurbinder|kaur.gaganpreet|kaur.gurjinder|kaur.gursharn|kaur.gurmeet|kaur.gurmanik|kaur.guneet|kaur.gurjeet"Sircar, S."https://zbmath.org/authors/?q=ai:sircar.s-k|sircar.sarthokSummary: We provide a comparison of the dispersion properties, specifically the time-amplification factor, the scaled group velocity, the error in the phase speed and scaled numerical diffusion coefficient of four spatiotemporal discretization schemes utilized for solving a linear, one-dimensional (1D) as well as a linear/nonlinear, two-dimensional (2D) advection diffusion reaction (ADR) equation: (a) Explicit \((\mathrm{RK}_2)\) temporal integration combined with the Optimal Upwind Compact Scheme (or OUCS3, [\textit{T. K. Sengupta} et al., J. Comput. Phys. 192, No. 2, 677--694 (2003; Zbl 1038.65082)]) and the central difference scheme \((\mathrm{CD}_2)\), (b) Fully implicit mid-point rule for time integration coupled with the OUCS3 and \textit{S. K. Lele}'s compact scheme [ibid. 103, No. 1, 16--42 (1992; Zbl 0759.65006)], (c) Implicit (mid-point rule)-explicit \((\mathrm{RK}_2)\) or IMEX time integration blended with OUCS3 and Lele's compact scheme (where the IMEX time integration follows the same ideology as introduced by \textit{U. M. Ascher} et al. [SIAM J. Numer. Anal. 32, No. 3, 797--823 (1995; Zbl 0841.65081)]), and (d) IMEX (mid-point/\(\mathrm{RK}_2)\) time integration melded with the New Combined Compact Difference scheme (or NCCD scheme, [\textit{T. K. Sengupta} et al., J. Comput. Phys. 228, No. 17, 6150--6168 (2009; Zbl 1173.76034)]). Analysis reveals the superior resolution features of the IMEX-OUCS3-Lele scheme and the IMEX-NCCD scheme including an enhanced region of asymptotic stability (a region with numerical amplification factor less than unity), a diminished region of spurious propagation characteristics (or a region of negative scaled group velocity) and a smaller region of nonzero phase speed error. In particular, the IMEX-NCCD scheme captures the correct propagation feature (or positive scaled group velocity) in the largest possible region in the wavenumber-Courant-Friedrichs-Lewy (CFL) number, parameter space, in comparison with the other three numerical methods. The in silico experiments investigating the role of \(q\)-waves in the numerical solution of the linear, 1D ADR equation divulge excellent Dispersion Relation Preservation (DRP) properties of the IMEX-NCCD scheme including minimal dissipation via implicit filtering and negligible unphysical oscillations (or Gibbs' phenomenon) on coarser grids. The numerical solution of the 2D viscous Burgers' Equation underline the supremacy of the IMEX method with regard to handling the `stiff' derivative terms in contrast with a fully explicit time integration method and lower computational time versus with a fully implicit scheme. The DRP resolution of the IMEX-NCCD scheme is further benchmarked by solving the classical two-dimensional (2D), Patlak-Keller-Segel (PKS) nonlinear parabolic model. Numerical results reveal that the spiky structure of the solution is oscillation free and, when compared with the Explicit-OUCS3-CD2 method, the solution is better resolved by the IMEX-NCCD method.An efficient numerical method for reaction-diffusion equation on the general curved surfaceshttps://zbmath.org/1502.650742023-02-24T16:48:17.026759Z"Song, Xin"https://zbmath.org/authors/?q=ai:song.xin"Li, Yibao"https://zbmath.org/authors/?q=ai:li.yibaoSummary: In this paper, we propose an efficient numerical algorithm for reaction-diffusion equation on the general curved surface. The surface is discretized by a mesh consisting of triangular grids. The partial differential operators are defined based on the surface mesh and its dual surface polygonal tessellation. The proposed method has three advantages including intrinsic geometry, conservation law, and convergence property. The proposed method only needs the information of \(1\)-ring of neighboring vertices for the divergence of a vector field and the Laplace-Beltrami operators, while the numerical conservation laws still hold. The proposed method avoids the global surface triangulation and its implementation is simple since we can explicitly define the Laplace-Beltrami operator by using the information of the neighborhood of each triangular grid. In order to obtain second-order temporal accuracy, we utilize the Crank-Nicolson formula to the reaction-diffusion system. The discrete system is solved by the biconjugate gradient stabilized method. The proposed algorithm is simple to implement and is second-order accurate both in space and time. Various numerical experiments are presented to demonstrate the efficiency of our algorithm.Error estimates of energy stable numerical schemes for Allen-Cahn equations with nonlocal constraintshttps://zbmath.org/1502.650752023-02-24T16:48:17.026759Z"Sun, Shouwen"https://zbmath.org/authors/?q=ai:sun.shouwen"Jing, Xiaobo"https://zbmath.org/authors/?q=ai:jing.xiaobo"Wang, Qi"https://zbmath.org/authors/?q=ai:wang.qiSummary: We present error estimates for four unconditionally energy stable numerical schemes developed for solving Allen-Cahn equations with nonlocal constraints. The schemes are linear and second order in time and space, designed based on the energy quadratization (EQ) or the scalar auxiliary variable (SAV) method, respectively. In addition to the rigorous error estimates for each scheme, we also show that the linear systems resulting from the energy stable numerical schemes are all uniquely solvable. Then, we present some numerical experiments to show the accuracy of the schemes, their volume-preserving as well as energy dissipation properties in a drop merging simulation.An optimal finite difference scheme with minimized dispersion and adaptive dissipation considering the spectral properties of the fully discrete schemehttps://zbmath.org/1502.650762023-02-24T16:48:17.026759Z"Sun, Zhensheng"https://zbmath.org/authors/?q=ai:sun.zhensheng"Hu, Yu"https://zbmath.org/authors/?q=ai:hu.yu"Ren, Yuxin"https://zbmath.org/authors/?q=ai:ren.yuxin"Mao, Kai"https://zbmath.org/authors/?q=ai:mao.kaiSummary: For the simulation of flow fields with a broad range of length scales, designing numerical schemes with good spectral properties is one of the most important issues. To improve the spectral properties of the semi-discrete finite different schemes, the authors have previously proposed the idea of optimizing the dispersion and dissipation properties separately and a class of finite difference scheme with minimized dispersion and controllable dissipation properties is thus developed [\textit{Z.-s. Sun} et al., J. Comput. Phys. 270, 238--254 (2014; Zbl 1349.76537)]. In the present paper, we further investigated this idea and extend it to the fully discrete scheme. In other words, the dispersion and dissipation errors induced by the temporal discretization are taken into consideration in the present paper. Moreover, a scale sensor is designed in the control of dissipation error to ensure the numerical scheme can automatically adjust its dissipation according to the local characteristic of the flow field. To achieve the shock-capturing capability, this optimized scheme is blended with the WENO-Z scheme to form a hybrid scheme. A number of benchmark test cases including the transportation of a linear wave, the propagation of a sound wave packet, the Shu-Osher problem, the double Mach reflection problem and the Rayleigh-Taylor instability problem are employed to verify the good spectral properties and robust shock-capturing capability of the proposed scheme.A new 6th-order WENO scheme with modified stencilshttps://zbmath.org/1502.650772023-02-24T16:48:17.026759Z"Wang, Yahui"https://zbmath.org/authors/?q=ai:wang.yahui"Du, Yulong"https://zbmath.org/authors/?q=ai:du.yulong"Zhao, Kunlei"https://zbmath.org/authors/?q=ai:zhao.kunlei"Yuan, Li"https://zbmath.org/authors/?q=ai:yuan.liSummary: In this article, a new 6th-order weighted essentially non-oscillatory (WENO) scheme is developed. As with previous 6th-order central-upwind WENO schemes, the present scheme is a convex combination of four candidate linear reconstructions. The difference is that the most upwind and downwind stencils use four cell values, while the inner two stencils nominally use three cell values but the original quadratic reconstructions are modified to be 4th-order approximations by adding cubic correction terms involving the five cell values of the classical 5th-order WENO scheme. Sixth-order accuracy of the new scheme in smooth regions including critical points is achieved by using a reference smoothness indicator. Several numerical examples show that the new scheme has higher resolution compared with the recently developed 6th-order WENO schemes.A high accuracy local one-dimensional explicit compact scheme for the 2D acoustic wave equationhttps://zbmath.org/1502.650782023-02-24T16:48:17.026759Z"Wu, Mengling"https://zbmath.org/authors/?q=ai:wu.mengling"Jiang, Yunzhi"https://zbmath.org/authors/?q=ai:jiang.yunzhi"Ge, Yongbin"https://zbmath.org/authors/?q=ai:ge.yongbin(no abstract)Numerical threshold of linearly implicit Euler method for nonlinear infection-age SIR modelshttps://zbmath.org/1502.650792023-02-24T16:48:17.026759Z"Yang, Huizi"https://zbmath.org/authors/?q=ai:yang.huizi"Yang, Zhanwen"https://zbmath.org/authors/?q=ai:yang.zhanwen"Liu, Shengqiang"https://zbmath.org/authors/?q=ai:liu.shengqiangSummary: In this paper, we consider a numerical threshold of a linearly implicit Euler method for a nonlinear infection-age SIR model. It is shown that the method shares the equilibria and basic reproduction number \(R_0\) of age-independent SIR models for any stepsize. Namely, the disease-free equilibrium is globally stable for numerical processes when \(R_0<1\) and the underlying endemic equilibrium is globally stable for numerical processes when \(R_0>1\). A natural extension to nonlinear infection-age models is presented with an initial mortality rate and the numerical thresholds, i.e., numerical basic reproduction numbers \(R^h\), are presented according to the infinite Leslie matrix. Although the numerical basic reproduction numbers \(R^h\) are not quadrature approximations to the exact threshold \(R_0\), the disease-free equilibrium is locally stable for numerical processes whenever \(R^h<1\). Moreover, a unique numerical endemic equilibrium exists for \(R^h>1\), which is locally stable for numerical processes. It is much more important that both the numerical thresholds and numerical endemic equilibria converge to the exact ones with accuracy of order 1. Therefore, the local dynamical behaviors of nonlinear infection-age models are visually displayed by the numerical processes. Finally, numerical applications to the influenza models are shown to illustrate our results.Isogeometric collocation discretizations for acoustic wave problemshttps://zbmath.org/1502.650802023-02-24T16:48:17.026759Z"Zampieri, Elena"https://zbmath.org/authors/?q=ai:zampieri.elena"Pavarino, Luca F."https://zbmath.org/authors/?q=ai:pavarino.luca-francoSummary: The acoustic wave problem is here discretized by collocation isogeometric analysis (IGA) in space and Newmark schemes of first and second order in time. A numerical study in the plane on both Cartesian and NURBS domains investigates the convergence rate of the proposed collocation Newmark-IGA method and its dependence on the main isogeometric parameters, the mesh size \(h\), the spline polynomial degree \(p\), the spline regularity \(k\), and on the time step size \(\Delta t\). In addition, a Ricker wavelet propagation test is accurately reproduced. The stability thresholds in time of the proposed method depend linearly on \(h\) and inversely on \(p\). Therefore, the proposed collocation Newmark-IGA method retains the good convergence and stability properties of standard Galerkin IGA and spectral element discretizations of acoustic problems, as well as the high computational efficiency of collocation methods due to matrix sparsity and fast function evaluation.Efficient, linear and fast numerical algorithm for the volume conserved nonlocal Allen-Cahn equationhttps://zbmath.org/1502.650812023-02-24T16:48:17.026759Z"Zeng, Shilin"https://zbmath.org/authors/?q=ai:zeng.shilin"Xie, Ziqing"https://zbmath.org/authors/?q=ai:xie.ziqing"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofeng"Wang, Jiangxing"https://zbmath.org/authors/?q=ai:wang.jiangxingSummary: In this article, for the conserved nonlocal Allen-Cahn equation, we construct a set of efficient, linear, unconditionally energy stable numerical schemes based on the Invariant Energy Quadratization approach. We not only show that the proposed schemes satisfy the law of energy dissipation unconditionally, but also derive the error estimates rigorously. Meanwhile, to reduce the computational cost and memory requirement caused by the nonlocal term, a fast implementation process based on the FFT technique is developed. Some numerical simulations are carried out to demonstrate the accuracy and stability of the schemes as well.Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regimehttps://zbmath.org/1502.650822023-02-24T16:48:17.026759Z"Zhang, Teng"https://zbmath.org/authors/?q=ai:zhang.teng"Wang, Tingchun"https://zbmath.org/authors/?q=ai:wang.tingchunSummary: We present rigorous analysis on the error bound and conservation laws of a fourth-order compact finite difference scheme for Zakharov system (ZS) with a dimensionless parameter \(\varepsilon \in (0,1]\), which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., \(0 < \varepsilon \ll 1\), the solutions have highly oscillatory waves and outgoing initial layers due to the perturbation from wave operator in ZS and the incompatibility of the initial data. The solutions propagate with \(O( \varepsilon)\) wavelength in time, \(O(1/ \varepsilon)\) speed in space, and \(O(\varepsilon^2)\) and \(O(1)\) amplitudes for well-prepared and ill-prepared initial data respectively. The high oscillation brings noticeable difficulties in analyzing the error bounds of numerical methods to the ZS. In this work, with \(h\) the mesh size and \(\tau\) the time step, we give a uniform error bound \(h^4+\tau^{2\alpha^{\dagger}/3}\) for the well- and less-ill-prepared initial data and an error bound \(h^4/ \varepsilon + \tau^2/ \varepsilon^3\) for the ill-prepared initial data with tools including energy methods and cut-off techniques. The compact scheme provides much better spatial resolution than general second-order methods and reduces the computational cost a lot. Numerical simulations are also provided to confirm our theoretical analysis.Asymptotically compatible energy law of the Crank-Nicolson type schemes for time-fractional MBE modelshttps://zbmath.org/1502.650832023-02-24T16:48:17.026759Z"Zhu, Xiaohan"https://zbmath.org/authors/?q=ai:zhu.xiaohan"Liao, Hong-lin"https://zbmath.org/authors/?q=ai:liao.honglinSummary: This paper is concerned with the numerical simulations for the dynamics of the time-fractional molecular beam epitaxy models. The variable-step Crank-Nicolson-type schemes are proposed and analyzed for model with and without slope selection respectively. By using the discrete gradient structures of discrete fractional derivative, the numerical schemes preserve discrete variational energy dissipation law on general meshes unconditionally. These are generalizations of classical energy dissipation laws and are asymptotically compatible with the classical MBE model. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our schemes.Conditions for \(L^2\)-dissipativity of an explicit symmetric finite-difference scheme for linearized 2D and 3D gas dynamics equations with a regularizationhttps://zbmath.org/1502.650842023-02-24T16:48:17.026759Z"Zlotnik, Alexander"https://zbmath.org/authors/?q=ai:zlotnik.alexander-aSummary: We study an explicit two-level in time and symmetric in space finite-difference scheme for a linearized 2D and 3D gas dynamic system of equations with a kinetic-type regularization. For an initial-boundary value problem on any nonuniform rectangular mesh, sufficient Courant-type conditions for the \(L^2\)-dissipativity are derived for the first time by the energy method. For the Cauchy problem on the uniform mesh, recent both necessary conditions and sufficient conditions for the \(L^2\)-dissipativity in the spectral method are improved. A new form for the relaxation parameter is suggested which guarantees that the Courant-type number is uniformly bounded from above and below with respect to the mesh and the Mach number.Convergence of a multi-point flux approximation finite volume scheme for a sharp-diffuse interfaces model for seawater intrusionhttps://zbmath.org/1502.650852023-02-24T16:48:17.026759Z"Amaziane, Brahim"https://zbmath.org/authors/?q=ai:amaziane.brahim"El Ossmani, Mustapha"https://zbmath.org/authors/?q=ai:el-ossmani.mustapha"Talali, Khadija"https://zbmath.org/authors/?q=ai:talali.khadijaSummary: This paper deals with development and analysis of a finite volume (FV) method for the coupled system describing seawater intrusion in coastal aquifers. The problem is modeled by the recently sharp-diffuse interfaces approach. This process is formulated by a coupled system of two nonlinear parabolic of cross-diffusion type equations describing two immiscible phase seawater/freshwater flow tacking into account the width of transition zones. A fully coupled, fully implicit approach is developed to discretize the coupled system. The method combines advantages of the MPFA method to accurately solve fluxes and diffusive terms and upstream for advective terms with implicit Euler's time discretization. The non-negativity of the discrete solution is proved and an existence result is shown using a fixed point theorem. Based on a priori estimates and compactness arguments, we prove the convergence of the numerical approximation to the weak solution. We have developed and implemented this scheme in a new module in the context of the open source platform DuMu\(^X\). Two numerical experiments are presented to demonstrate the efficiency of this scheme, one of which is related to flows in a fractured porous aquifer.Stability and convergence of spatial discrete stabilized finite volume method for the unsteady incompressible magnetohydrodynamics equationshttps://zbmath.org/1502.650862023-02-24T16:48:17.026759Z"Chu, Xiaochen"https://zbmath.org/authors/?q=ai:chu.xiaochen"Chen, Chuanjun"https://zbmath.org/authors/?q=ai:chen.chuanjun"Zhang, Tong"https://zbmath.org/authors/?q=ai:zhang.tong.1|zhang.tong|zhang.tong.3|zhang.tong.2Summary: In this paper, the stability and convergence results of finite volume method for the time-dependent incompressible magnetohydrodynamics equations are developed based on triangulation. The linear polynomials are used to approximate the velocity, magnetic fields and pressure. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection method is employed. Firstly, two self-adjoint linear mappings are introduced to establish the equivalences of temporal terms and bilinear terms in finite element method and finite volume method. Secondly, the existence and uniqueness of numerical solutions are presented by using the Oseen iteration and mathematical induction method. Thirdly, \( H^2\)-stability and optimal error estimates are provided by introducing the Galerkin projection of generalized bilinear terms in finite volume method, using the energy method and constructing the corresponding dual problem. The main theoretical findings of this paper are as follows:
\[
\| ( u_h ( t ) , H_h ( t ) ) \|_0^2 + \int_0^t(\min \{\nu, \gamma \sigma^{- 1} \} \| ( u_h , H_h ) \|_1^2 + G( p_h, p_h))\, d s \leq C,
\]
\[\min \{\nu, \gamma \sigma^{- 1} \} \| ( u_h ( t ) , H_h ( t ) ) \|_1^2 + G( p_h(t), p_h(t)) + \int_0^t \| ( u_{h t} , H_{h t} ) \|_0^2\, d s \leq C,
\]
\[
\|\nabla ( u_h , H_h ) \|_0^2 + \int_0^t \min \{\nu, \gamma \sigma^{- 1} \} \| ( A_{1 h} u_h , A_{2 h} H_h ) \|_0^2\, d s \leq C,
\]
\[
\tau^{\frac{ 1}{ 2}}(t) \| ( u ( t ) - u_h ( t ) , H ( t ) - H_h ( t ) ) \|_0 + h( \| ( u ( t ) - u_h ( t ) , H ( t ) - H_h ( t ) ) \|_1 + \tau^{\frac{ 1}{ 2}}(t) \| p ( t ) - p_h ( t ) \|_0) \leq C h^2,
\]
where \(\tau(t) = \min \{1, t \} \), the letters \(\nu, \sigma\) are the physical parameters, \(C\) and \(\gamma\) are two positive constants independent of the mesh size \(h, (u, H, p)\) and \(( u_h, H_h, p_h)\) are the solutions of problems (2.1) and (4.1). Finally, some numerical results are presented to verify the established theoretical findings and show the performances of the considered numerical method.A hybrid reduced order method for modelling turbulent heat transfer problemshttps://zbmath.org/1502.650872023-02-24T16:48:17.026759Z"Georgaka, Sokratia"https://zbmath.org/authors/?q=ai:georgaka.sokratia"Stabile, Giovanni"https://zbmath.org/authors/?q=ai:stabile.giovanni"Star, Kelbij"https://zbmath.org/authors/?q=ai:star.kelbij"Rozza, Gianluigi"https://zbmath.org/authors/?q=ai:rozza.gianluigi"Bluck, Michael J."https://zbmath.org/authors/?q=ai:bluck.michael-jSummary: A parametric, hybrid reduced order method based on the Proper Orthogonal Decomposition with both Galerkin projection and interpolation based on Radial Basis Functions method is presented. This method is tested on a case of turbulent non-isothermal mixing in a T-junction pipe, a common flow arrangement found in nuclear reactor cooling systems. The reduced order model is derived from the 3D unsteady, incompressible Navier-Stokes equations weakly coupled with the energy equation. For high Reynolds numbers, the eddy viscosity and eddy diffusivity are incorporated into the Reduced Order Model with a Proper Orthogonal Decomposition (nested and standard) with Interpolation (PODI), where the interpolation is performed using Radial Basis Functions. The reduced order solver, obtained using a \(k - \omega\) SST Unsteady Reynolds Averaged Navier-Stokes full order model, is tested against the full order solver in a 3D T-junction pipe with parameterised velocity inlet boundary conditions.A novel large eddy simulation model for the quasi-geostrophic equations in a finite volume settinghttps://zbmath.org/1502.650882023-02-24T16:48:17.026759Z"Girfoglio, Michele"https://zbmath.org/authors/?q=ai:girfoglio.michele"Quaini, Annalisa"https://zbmath.org/authors/?q=ai:quaini.annalisa"Rozza, Gianluigi"https://zbmath.org/authors/?q=ai:rozza.gianluigiSummary: We present a Large Eddy Simulation (LES) approach based on a nonlinear differential low-pass filter for the simulation of two-dimensional barotropic flows with under-refined meshes. For the implementation of such model, we choose a segregated three-step algorithm combined with a computationally efficient Finite Volume method. We assess the performance of our approach with the classical double-gyre wind forcing benchmark. The numerical experiments we present demonstrate that our nonlinear filter is an improvement over a linear filter since it is able to recover the four-gyre pattern of the time-averaged stream function even with extremely coarse meshes. In addition, our LES approach provides an average kinetic energy that compares well with the one computed with a Direct Numerical Simulation.Sensitivity parameter-independent characteristic-wise well-balanced finite volume WENO scheme for the Euler equations under gravitational fieldshttps://zbmath.org/1502.650892023-02-24T16:48:17.026759Z"Li, Peng"https://zbmath.org/authors/?q=ai:li.peng"Wang, Bao-Shan"https://zbmath.org/authors/?q=ai:wang.baoshan"Don, Wai-Sun"https://zbmath.org/authors/?q=ai:don.wai-sunSummary: Euler equations with a gravitational source term (PDEs) admit a hydrostatic equilibrium state where the source term exactly balances the flux gradient. The property of exact preservation of the equilibria is highly desirable when the PDEs are numerically solved. \textit{G. Li} and \textit{Y. Xing} [J. Comput. Phys. 316, 145--163 (2016; Zbl 1349.76356)] proposed a high-order well-balanced characteristic-wise finite volume weighted essentially non-oscillatory (FV-WENO) scheme for the cases of isothermal equilibrium and polytropic equilibrium. On the contrary to what was claimed, the scheme is not well-balanced. The root of the problem is the precarious effects of a non-zero sensitivity parameter in the nonlinear weights of the WENO polynomial reconstruction procedure (WENO operator). The effects are identified in the theoretical proof for the well-balanced scheme and verified numerically on a coarse mesh resolution and a long time simulation of the PDEs. In this study, two simple yet effective numerical techniques derived from the multiplicative-invariance (MI) property of a WENO operator are invoked to rectify the sensitivity parameter's dependency yielding a correct proof for the sensitivity parameter-independent (characteristic-wise) well-balanced FV-WENO scheme. The (non-)well-balanced nature of the schemes is demonstrated with several one- and two-dimensional benchmark steady state problems and a small perturbation over the steady state problems. Moreover, the one-dimensional Sod problem under the gravitational field is also simulated for showing the performance of the well-balanced FV-WENO scheme in capturing shock, contact discontinuity, and rarefaction wave in an essentially non-oscillatory nature. It also indicates that the numerical scheme with the third-order Runge-Kutta time-stepping scheme should take the CFL number less than 0.5 to mitigate the Gibbs oscillations at the shock without increasing the numerical dissipation artificially in the Lax-Friedrichs numerical flux.Affine-invariant WENO weights and operatorhttps://zbmath.org/1502.650902023-02-24T16:48:17.026759Z"Wang, Bao-Shan"https://zbmath.org/authors/?q=ai:wang.baoshan"Don, Wai Sun"https://zbmath.org/authors/?q=ai:don.wai-sunSummary: The novel and simple nonlinear affine-invariant weights (Ai-weights) are devised for the Ai-WENO operator to handle the case when the function being reconstructed undergoes an affine transformation (Ai-operator) with a constant scaling and translation (Ai-coefficients) within a global (WENO) stencil. The Ai-weights essentially decouple the inter-dependencies of the Ai-coefficients and sensitivity parameter effectively. For any given sensitivity parameter, the Ai-WENO operator guarantees that the WENO-reconstructed affine-transformed-function remains unchanged as the affine-transformed WENO-reconstructed-function. In other words, the Ai-operator commutes with the nonlinear WENO operator (Ai-property) as proven theoretically and validated numerically. With a small scaling and a non-zero translation, the Ai-WENO scheme with a typical sensitivity parameter satisfies the ENO-property even when the corresponding classical and scale-invariant WENO scheme does not. In solving the shallow water wave equations and the Euler equations under gravitational fields, the characteristic-wise well-balanced Ai-WENO scheme satisfies the well-balanced property intrinsically without imposing the WENO linearization technique. Any Ai-weights-based WENO operator enhances the robustness and reliability of the WENO scheme for solving hyperbolic conservation laws.Construction of nonlinear approximation schemes for piecewise smooth datahttps://zbmath.org/1502.650912023-02-24T16:48:17.026759Z"Yang, Hyoseon"https://zbmath.org/authors/?q=ai:yang.hyoseon"Yoon, Jungho"https://zbmath.org/authors/?q=ai:yoon.junghoSummary: The objective of this paper is to develop a family of nonlinear approximation schemes for piecewise smooth data on \(\mathbb R\). The quasi-interpolation method is a very efficient tool for reconstructing functions from a discrete set of their values on \(\mathbb R\). It has advantages in simplicity and fast computation. However, when approximating near singularities or sharp gradients of the underlying functions, it often suffers from spurious oscillations or blurring edges. Motivated by this observation, we present a nonlinear modification of the quasi-interpolation to prevent such undesirable artifacts near singular points, while achieving high-order accuracy in smooth regions. To this end, we first introduce two important tools, a smoothness indicator and a singularity detector, and then construct new nonlinear kernels. A detailed error analysis of the proposed scheme is provided. We show that, in smooth regions, the proposed scheme achieves the same approximation order as its linear counterpart, while maintaining essentially non-oscillatory behavior near the singularities. Finally, some numerical results are presented to demonstrate the ability of the proposed nonlinear scheme.Error identities for the reaction-convection-diffusion problem and applications to a posteriori error controlhttps://zbmath.org/1502.650922023-02-24T16:48:17.026759Z"Repin, Sergey I."https://zbmath.org/authors/?q=ai:repin.sergey-iSummary: The paper is devoted to a posteriori error identities for the stationary reaction-convection-diffusion problem with mixed Dirichlét-Neumann boundary conditions. They reflect the most general relations between deviations of approximations from the exact solutions and those values that can be observed in a numerical experiment. The identities contain no mesh dependent constants and are valid for any function in the admissible (energy) class. Therefore, the identities and the estimates that follow from them generate universal and fully reliable tools of a posteriori error control.PiTSBiCG: parallel in time stable bi-conjugate gradient algorithmhttps://zbmath.org/1502.650932023-02-24T16:48:17.026759Z"Riahi, Mohamed Kamel"https://zbmath.org/authors/?q=ai:riahi.mohamed-kamelSummary: This paper presents a new algorithm for the parallel in time (PiT) numerical simulation of time dependent partial/ordinary differential equations. We propose a reliable alternative to the well know parareal in time algorithm, by formulating the parallel in time problem algebraically and solve it using an adapted Bi-Conjugate gradient stabilized method. The proposed Parallel in time Stable Bi-Conjugate algorithm (PiTSBiCG for short) has a great potential to stabilizing the parallel resolution for a variety of problems. In this work, we describe the mathematical approach to the new algorithm and provide numerical evidence that shows its superiority to the standard parareal method.Discrete mollification in Bernstein basis and space marching scheme for numerical solution of an inverse two-phase one-dimensional Stefan problemhttps://zbmath.org/1502.650942023-02-24T16:48:17.026759Z"Bodaghi, S."https://zbmath.org/authors/?q=ai:bodaghi.soheila"Zakeri, A."https://zbmath.org/authors/?q=ai:zakeri.ali"Amiraslani, A."https://zbmath.org/authors/?q=ai:amiraslani.amir"Shayegan, A. H. Salehi"https://zbmath.org/authors/?q=ai:shayegan.amir-hossein-salehiSummary: In this paper, a one-dimensional two-phase inverse Stefan problem is studied. The free surface is considered unknown here, which is more realistic from the practical point of view. The problem is ill-posed since small errors in the input data can lead to large deviations from the desired solution. To obtain a stable numerical solution, we propose a method based on discrete mollification combined with space marching. We use the integration matrix in Bernstein polynomial basis for the discrete mollification method. Through this method, our numerical integration does not involve any direct function integration and only contains algebraic calculations. Furthermore, the stability and convergence of the process are proved. Finally, the results of this paper are illustrated and examined through some numerical examples. The numerical examples supporting our theoretical analysis are provided and discussed.On the numerical solutions of some identification problems for one- and multidimensional parabolic equations backward in timehttps://zbmath.org/1502.650952023-02-24T16:48:17.026759Z"Sazaklioglu, Ali Ugur"https://zbmath.org/authors/?q=ai:sazaklioglu.ali-ugurSummary: In this paper, an abstract (differential) inverse problem backward in time, that is a class of some identification problems for simultaneous determination of the source and initial condition, is considered. A finite difference scheme is constructed for the numerical solution of this abstract problem. Some stability and almost coercive stability estimates for the constructed difference scheme are established by the tools of operator theory. Furthermore, the proposed abstract difference scheme and the acquired results for that are extended for several applications involving identification problems for one- and multidimensional parabolic equations. Finally, sundry illustrative numerical results and visualizations are carried out and discussed.Numerical analysis of a Darcy-Forchheimer model coupled with the heat equationhttps://zbmath.org/1502.650962023-02-24T16:48:17.026759Z"Deugoue, G."https://zbmath.org/authors/?q=ai:deugoue.gabriel"Djoko, J. K."https://zbmath.org/authors/?q=ai:djoko.jules-k"Konlack, V. S."https://zbmath.org/authors/?q=ai:konlack.virginie-s"Mbehou, M."https://zbmath.org/authors/?q=ai:mbehou.mohamedSummary: This paper discusses a novel three field formulation for the Darcy-Forchheimer flow with a nonlinear viscosity depending on the temperature coupled with the heat equation. We show unique solvability of the variational problem by using; Galerkin method, Brouwer's fixed point and some compactness properties. We propose and study in detail a finite element approximation. A priori error estimate is then derived and convergence is obtained. A solution technique is formulated to solve the nonlinear problem and numerical experiments that validate the theoretical findings are presented.Towards a time adaptive Neumann-Neumann waveform relaxation method for thermal fluid-structure interactionhttps://zbmath.org/1502.650972023-02-24T16:48:17.026759Z"Monge, Azahar"https://zbmath.org/authors/?q=ai:monge.azahar"Birken, Philipp"https://zbmath.org/authors/?q=ai:birken.philippSummary: Our prime motivation is thermal fluid-structure interaction (FSI) where two domains with jumps in the material coefficients are connected through an interface. There exist two main strategies to simulate FSI models: the monolithic approach where a new code is tailored for the coupled equations and the partitioned approach that allows to reuse existing software for each sub-problem. Here we want to develop multirate methods that contribute to the time parallelization of the sub-problems for the partitioned simulation of FSI problems.
For the entire collection see [Zbl 1485.65003].On one approach for the numerical solving of hyperbolic initial-boundary problems on an adaptive moving gridshttps://zbmath.org/1502.650982023-02-24T16:48:17.026759Z"Shilnikov, K. E."https://zbmath.org/authors/?q=ai:shilnikov.kirill-e"Kochanov, M. B."https://zbmath.org/authors/?q=ai:kochanov.mark-bSummary: An approach for the quality improvement of numerical modeling of the processes described by nonlinear hyperbolic equations, which have conservative form is proposed. The transition to synchronous moving coordinate system along with Godunov-type numerical scheme for the problem on moving mesh is applied. The grid motion law relies heavily on Rankine-Hugoniot relations. In order to prevent the degeneration of moving computational mesh the regularization mechanism is implemented. It allows to almost automatically fit the computational mesh to the admissible width of the solution singularity resolution. The proposed algorithm is tested on several initial-boundary problems for the Hopf equation with well known exact solutions. The hyperbolic Buckley-Leverett equation is also considered as an example of application of the proposed approach.Multi-step variant of the parareal algorithmhttps://zbmath.org/1502.650992023-02-24T16:48:17.026759Z"Ait-Ameur, Katia"https://zbmath.org/authors/?q=ai:ait-ameur.katia"Maday, Yvon"https://zbmath.org/authors/?q=ai:maday.yvon"Tajchman, Marc"https://zbmath.org/authors/?q=ai:tajchman.marcSummary: In the field of nuclear energy, computations of complex two-phase flows are required for the design and safety studies of nuclear reactors. System codes are dedicated to the thermal-hydraulic analysis of nuclear reactors at system scale by simulating the whole reactor.
For the entire collection see [Zbl 1485.65003].Planar multi-patch domain parameterization for isogeometric analysis based on evolution of fat skeletonhttps://zbmath.org/1502.651002023-02-24T16:48:17.026759Z"Bastl, Bohumír"https://zbmath.org/authors/?q=ai:bastl.bohumir"Slabá, Kristýna"https://zbmath.org/authors/?q=ai:slaba.kristynaSummary: In this paper, we propose a new algorithm for computing planar multi-patch domain parameterizations which are suitable for consequent numerical computations based on isogeometric analysis. Firstly, a so-called fat skeleton of the given domain, represented by a multi-patch NURBS parameterization, is constructed, which determines the topology of a multi-patch NURBS parameterization suitable for the given domain. Then, we formulate an evolution process which successively transforms the initial shape represented by the fat skeleton to the target shape represented by the given domain. Moreover, a multi-patch optimization is applied in every time iteration to make the multi-patch NURBS parameterization as uniform and orthogonal as possible. The method is constructed such that it is fully automatic and it solves the biggest problems of the domain parameterization problem, i.e., it automatically determines a suitable topology of a multi-patch NURBS parameterization for the given domain and it also automatically solves the boundary correspondence problem. The functionality of the proposed method is demonstrated on several examples.Newton-Krylov-BDDC deluxe solvers for non-symmetric fully implicit time discretizations of the bidomain modelhttps://zbmath.org/1502.651012023-02-24T16:48:17.026759Z"Huynh, Ngoc Mai Monica"https://zbmath.org/authors/?q=ai:monica-huynh.ngoc-maiSummary: A novel theoretical convergence rate estimate for a Balancing Domain Decomposition by Constraints algorithm is proven for the solution of the cardiac bidomain model, describing the propagation of the electric impulse in the cardiac tissue. The non-linear system arises from a fully implicit time discretization and a monolithic solution approach. The preconditioned non-symmetric operator is constructed from the linearized system arising within the Newton-Krylov approach for the solution of the non-linear problem; we theoretically analyze and prove a convergence rate bound for the Generalised Minimal Residual iterations' residual. The theory is confirmed by extensive parallel numerical tests, widening the class of robust and efficient solvers for implicit time discretizations of the bidomain model.BDDC preconditioners for a space-time finite element discretization of parabolic problemshttps://zbmath.org/1502.651022023-02-24T16:48:17.026759Z"Langer, Ulrich"https://zbmath.org/authors/?q=ai:langer.ulrich"Yang, Huidong"https://zbmath.org/authors/?q=ai:yang.huidongSummary: Continuous space-time finite element methods for parabolic problems have been recently studied, e.g., in [\textit{R. E. Bank} et al., J. Comput. Appl. Math. 310, 19--31 (2017; Zbl 1348.65139); \textit{U. Langer} et al., Comput. Methods Appl. Mech. Eng. 306, 342--363 (2016; Zbl 1436.76027); \textit{U. Langer} et al., Lect. Notes Comput. Sci. Eng. 128, 247--275 (2019; Zbl 1433.65217); \textit{O. Steinbach}, Comput. Methods Appl. Math. 15, No. 4, 551--566 (2015; Zbl 1329.65229)]. The main common features of these methods are very different from those of time-stepping methods. Time is considered to be just another spatial coordinate. The variational formulations are studied in the full spacetime cylinder that is then decomposed into arbitrary admissible simplex elements.
For the entire collection see [Zbl 1485.65003].Local and parallel finite element methods based on two-grid discretizations for the nonstationary Navier-Stokes equationshttps://zbmath.org/1502.651032023-02-24T16:48:17.026759Z"Li, Qingtao"https://zbmath.org/authors/?q=ai:li.qingtao"Du, Guangzhi"https://zbmath.org/authors/?q=ai:du.guangzhiSummary: In this paper, some local and parallel finite element methods based on two-grid discretizations are proposed and investigated for the unsteady Navier-Stokes equations. The backward Euler scheme is considered for the temporal discretization, and two-grid method is used for the space discretization. The key idea is that for a solution to the unsteady Navier-Stokes problem, we could use a relatively coarse mesh to approximate low-frequency components and use some local fine mesh to compute high-frequency components. Some local a priori estimate is obtained. With that, theoretical results are derived. Finally, some numerical results are reported to support the theoretical findings.Fourier analysis of a time-simultaneous two-grid algorithm using a damped Jacobi waveform relaxation smoother for the one-dimensional heat equationhttps://zbmath.org/1502.651042023-02-24T16:48:17.026759Z"Lohmann, Christoph"https://zbmath.org/authors/?q=ai:lohmann.christoph"Dünnebacke, Jonas"https://zbmath.org/authors/?q=ai:dunnebacke.jonas"Turek, Stefan"https://zbmath.org/authors/?q=ai:turek.stefanSummary: In this work, the convergence behavior of a time-simultaneous two-grid algorithm for the one-dimensional heat equation is studied using Fourier arguments in space. The underlying linear system of equations is obtained by a finite element or finite difference approximation in space while the semi-discrete problem is discretized in time using the \(\vartheta\)-scheme. The simultaneous treatment of all time instances leads to a global system of linear equations which provides the potential for a higher degree of parallelization of multigrid solvers due to the increased number of degrees of freedom per spatial unknown. It is shown that the all-at-once system based on an equidistant discretization in space and time stays well conditioned even if the number of blocked time-steps grows arbitrarily. Furthermore, mesh-independent convergence rates of the considered two-grid algorithm are proved by adopting classical Fourier arguments in space without assuming periodic boundary conditions. The rate of convergence with respect to the Euclidean norm does not deteriorate arbitrarily if the number of blocked time steps increases and, hence, underlines the potential of the solution algorithm under investigation. Numerical studies demonstrate why minimizing the spectral norm of the iteration matrix may be practically more relevant than improving the asymptotic rate of convergence.Two-grid domain decomposition methods for the coupled Stokes-Darcy systemhttps://zbmath.org/1502.651052023-02-24T16:48:17.026759Z"Sun, Yizhong"https://zbmath.org/authors/?q=ai:sun.yizhong"Shi, Feng"https://zbmath.org/authors/?q=ai:shi.feng"Zheng, Haibiao"https://zbmath.org/authors/?q=ai:zheng.haibiao"Li, Heng"https://zbmath.org/authors/?q=ai:li.heng"Wang, Fan"https://zbmath.org/authors/?q=ai:wang.fanSummary: In this paper, we propose two novel Robin-type domain decomposition methods based on the two-grid techniques for the coupled Stokes-Darcy system. Our schemes firstly adopt the existing Robin-type domain decomposition algorithm to obtain the coarse grid approximate solutions. Then two one-step modified domain decomposition methods are further constructed on the fine grid by utilizing the framework of two-grid methods to enhance computational efficiency, via replacing some interface terms with the coarse grid information. The natural idea of using the two-grid frame to optimize the domain decomposition method inherits the best features of both methods and can overcome some of the domain decomposition deficits. The resulting schemes can be implemented easily using many existing mature solvers or codes in a flexible way, which are much effective under smaller mesh sizes or some realistic physical parameters. Moreover, several error estimates are carried out to show the stability and convergence of the schemes. Finally, three numerical experiments are performed and compared with the classical two-grid method, which verifies the validation and efficiency of the proposed algorithms.Coupling of Navier-Stokes equations and their hydrostatic versions for ocean flows: a discussion on algorithm and implementationhttps://zbmath.org/1502.651062023-02-24T16:48:17.026759Z"Tang, Hansong"https://zbmath.org/authors/?q=ai:tang.hansong"Liu, Yingjie"https://zbmath.org/authors/?q=ai:liu.yingjieSummary: Now it is necessary to advance our capabilities to direct simulation of many emerging problems of coastal ocean flows. Two examples of such flow problems are the 2010 Gulf of Mexico oil spill and the 2011 Japan tsunami. The two examples come from different backgrounds, however, they present a same challenge to our modeling capacity; the both examples involve distinct types of physical phenomena at vastly different scales, and they are multiscale and multiphysics flows in nature.
For the entire collection see [Zbl 1485.65003].Error analysis of a class of semi-discrete schemes for solving the Gross-Pitaevskii equation at low regularityhttps://zbmath.org/1502.651072023-02-24T16:48:17.026759Z"Alama, Yvonne Bronsard"https://zbmath.org/authors/?q=ai:alama.yvonne-bronsardSummary: We analyze a class of time discretizations for solving the nonlinear Schrödinger equation with non-smooth potential and at low-regularity on an arbitrary Lipschitz domain \(\Omega \subset \mathbb{R}^d\), \(d \leq 3\). We show that these schemes, together with their optimal local error structure, allow for convergence under lower regularity assumptions on both the solution and the potential than is required by classical methods, such as splitting or exponential integrator methods. Moreover, we show first and second order convergence in the case of periodic boundary conditions, in any fractional positive Sobolev space \(H^r\), \(r \geq 0\), beyond the more typical \(L^2\) or \(H^\sigma\) (\(\sigma > \frac{d}{2}\))-error analysis. Numerical experiments illustrate our results.Analysis of backward Euler projection FEM for the Landau-Lifshitz equationhttps://zbmath.org/1502.651082023-02-24T16:48:17.026759Z"An, Rong"https://zbmath.org/authors/?q=ai:an.rong"Sun, Weiwei"https://zbmath.org/authors/?q=ai:sun.weiweiSummary: The paper focuses on the analysis of the Euler projection Galerkin finite element method (FEM) for the dynamics of magnetization in ferromagnetic materials, described by the Landau-Lifshitz equation with the point-wise constraint \(|\mathbf{m}| = 1\). The method is based on a simple sphere projection that projects the numerical solution onto a unit sphere at each time step, and the method has been used in many areas in the past several decades. However, error analysis for the commonly used method has not been done since the classical energy approach cannot be applied directly. In this paper we present an optimal \(\mathbf{L}^2\) error analysis of the backward Euler sphere projection method by using quadratic or higher order finite elements under a time step condition \(\tau = O(\epsilon_0 h)\) with some small \(\epsilon_0 > 0\). The analysis is based on more precise estimates of the extra error caused by the sphere projection in both \(\mathbf{L}^2\) and \(\mathbf{H}^1\) norms, and the classical estimate of dual norm. Numerical experiment is provided to confirm our theoretical analysis.Galerkin approximation for one-dimensional wave equation by quadratic B-splineshttps://zbmath.org/1502.651092023-02-24T16:48:17.026759Z"Arar, Nouria"https://zbmath.org/authors/?q=ai:arar.nouriaSummary: This work is devoted to the development of a Galerkin-type approximation of the solution of a wave equation, using quadratic B-Spline functions and a 2-centred finite difference scheme. Two examples are used to validate the proposed approximation. The numerical results obtained show the effectiveness of the procedure for very small times. This makes it attractive for the approximation of PDEs with not known explicit solution.Numerical analysis of a wave equation for lossy media obeying a frequency power lawhttps://zbmath.org/1502.651102023-02-24T16:48:17.026759Z"Baker, Katherine"https://zbmath.org/authors/?q=ai:baker.katherine"Banjai, Lehel"https://zbmath.org/authors/?q=ai:banjai.lehelSummary: We study a wave equation with a nonlocal time fractional damping term that models the effects of acoustic attenuation characterized by a frequency-dependent power law. First we prove the existence of a unique solution to this equation with particular attention paid to the handling of the fractional derivative. Then we derive an explicit time-stepping scheme based on the finite element method in space and a combination of convolution quadrature and second-order central differences in time. We conduct a full error analysis of the mixed time discretization and in turn the fully space-time discretized scheme. Error estimates are given for both smooth solutions and solutions with a singularity at \(t = 0\) of a type that is typical for equations involving fractional time derivatives. A number of numerical results are presented to support the error analysis.Numerical approximation of a system of Hamilton-Jacobi-Bellman equations arising in innovation dynamicshttps://zbmath.org/1502.651112023-02-24T16:48:17.026759Z"Baňas, Ľubomír"https://zbmath.org/authors/?q=ai:banas.lubomir"Dawid, Herbert"https://zbmath.org/authors/?q=ai:dawid.herbert"Randrianasolo, Tsiry Avisoa"https://zbmath.org/authors/?q=ai:randrianasolo.tsiry-avisoa"Storn, Johannes"https://zbmath.org/authors/?q=ai:storn.johannes"Wen, Xingang"https://zbmath.org/authors/?q=ai:wen.xingangIn this paper, different discretization methods are compared for the approximation of Hamilton-Jacobi-Bellman equations with the value functions of an optimal innovation investment problem of a monopoly firm facing bankruptcy risk. The authors consider a stabilized finite element method (FEM), two adaptive finite element discretization strategies; the first approach employs a stabilized finite element approximation and the second one is based on an adaptive least-squares finite element method (LSFEM). The proposed FEMs are compared with alternative discretization approaches: the Chebyshev collocation method and the finite difference method. The implementation of the considered numerical scheme is discussed in detail such as the convergence of the policy iteration steady-state and time-dependent problems. The performance of the methods is discussed on several numerical examples with different model parameters.
Reviewer: Bülent Karasözen (Ankara)Continuous/discontinuous Galerkin difference discretizations of high-order differential operatorshttps://zbmath.org/1502.651122023-02-24T16:48:17.026759Z"Banks, J. W."https://zbmath.org/authors/?q=ai:banks.jeffrey-w"Buckner, B. Brett"https://zbmath.org/authors/?q=ai:buckner.benjamin-brett"Hagstrom, T."https://zbmath.org/authors/?q=ai:hagstrom.thomas-mSummary: We develop continuous/discontinuous discretizations for high-order differential operators using the Galerkin Difference approach. Grid dispersion analyses are performed that indicate a nodal superconvergence in the \(\ell^2\) norm. A treatment of the boundary conditions is described that ultimately leads to moderate growth in the spectral radius of the operators with polynomial degree, and in general the norms of the Galerkin Difference differentiation operators are significantly smaller than those arising from standard elements. Lastly, we observe that with the use of the Galerkin Difference space, the standard penalty terms required for discretizing high-order operators are not needed. Numerical results confirm the conclusions of the analyses performed.Simulation of constrained elastic curves and application to a conical sheet indentation problemhttps://zbmath.org/1502.651132023-02-24T16:48:17.026759Z"Bartels, Sören"https://zbmath.org/authors/?q=ai:bartels.sorenSummary: We consider variational problems that model the bending behavior of curves that are constrained to belong to given hypersurfaces. Finite element discretizations of corresponding functionals are justified rigorously via \(\varGamma\)-convergence. The stability of semi-implicit discretizations of gradient flows is investigated, which provide a practical method to determine stationary configurations. A particular application of the considered models arises in the description of conical sheet deformations.Backward Euler method for the equations of motion arising in Oldroyd model of order one with nonsmooth initial datahttps://zbmath.org/1502.651142023-02-24T16:48:17.026759Z"Bir, Bikram"https://zbmath.org/authors/?q=ai:bir.bikram"Goswami, Deepjyoti"https://zbmath.org/authors/?q=ai:goswami.deepjyoti"Pani, Amiya K."https://zbmath.org/authors/?q=ai:pani.amiya-kumarSummary: In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed for the equations of motion arising in the two-dimensional Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in \(L^\infty\) in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal \textit{a priori} error estimate in \(\mathbf{L}^2\)-norm is derived for the discrete problem with nonsmooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition. Finally, we present some numerical results to validate our theoretical results.Mortar-based entropy-stable discontinuous Galerkin methods on non-conforming quadrilateral and hexahedral mesheshttps://zbmath.org/1502.651152023-02-24T16:48:17.026759Z"Chan, Jesse"https://zbmath.org/authors/?q=ai:chan.jesse"Bencomo, Mario J."https://zbmath.org/authors/?q=ai:bencomo.mario-j"Del Rey Fernández, David C."https://zbmath.org/authors/?q=ai:del-rey-fernandez.david-cSummary: High-order entropy-stable discontinuous Galerkin (DG) methods for nonlinear conservation laws reproduce a discrete entropy inequality by combining entropy conservative finite volume fluxes with summation-by-parts (SBP) discretization matrices. In the DG context, on tensor product (quadrilateral and hexahedral) elements, SBP matrices are typically constructed by collocating at Lobatto quadrature points. Recent work has extended the construction of entropy-stable DG schemes to collocation at more accurate Gauss quadrature points [\textit{J. Chan} et al., SIAM J. Sci. Comput. 41, No. 5, A2938--A2966 (2019; Zbl 1435.65172)]. In this work, we extend entropy-stable Gauss collocation schemes to non-conforming meshes. Entropy-stable DG schemes require computing entropy conservative numerical fluxes between volume and surface quadrature nodes. On conforming tensor product meshes where volume and surface nodes are aligned, flux evaluations are required only between ``lines'' of nodes. However, on non-conforming meshes, volume and surface nodes are no longer aligned, resulting in a larger number of flux evaluations. We reduce this expense by introducing an entropy-stable mortar-based treatment of non-conforming interfaces via a face-local correction term, and provide necessary conditions for high-order accuracy. Numerical experiments for the compressible Euler equations in two and three dimensions confirm the stability and accuracy of this approach.The study of the numerical treatment of the real Ginzburg-Landau equation using the Galerkin methodhttps://zbmath.org/1502.651162023-02-24T16:48:17.026759Z"Chin, Pius W. M."https://zbmath.org/authors/?q=ai:chin.pius-w-mSummary: The real Ginzburg-Landau equation is studied in this article. We proceed and use the Galerkin method in combination with the compactness theorem to show that the solution of this problem exists and is unique in appropriate Sobolev spaces. This is proceeded by the designing of a reliable nonlinear numerical scheme from the afore mentioned problem and further show that this scheme is stable. Furthermore, the optimal rate of convergence of the scheme is determined in some appropriate spaces with emphasizes on the fact that the numerical solution from this scheme preserves all the qualitative properties of the exact solution and the numerical experiments are conducted with the help of an example to justify the theory.Bernstein-Bézier Galerkin-characteristics finite element method for convection-diffusion problemshttps://zbmath.org/1502.651172023-02-24T16:48:17.026759Z"El-Amrani, Mofdi"https://zbmath.org/authors/?q=ai:el-amrani.mofdi"El-Kacimi, Abdellah"https://zbmath.org/authors/?q=ai:el-kacimi.abdellah"Khouya, Bassou"https://zbmath.org/authors/?q=ai:khouya.bassou"Seaid, Mohammed"https://zbmath.org/authors/?q=ai:seaid.mohammedSummary: A class of Bernstein-Bézier basis based high-order finite element methods is developed for the Galerkin-characteristics solution of convection-diffusion problems. The Galerkin-characteristics formulation is derived using a semi-Lagrangian discretization of the total derivative in the considered problems. The spatial discretization is performed using the finite element method on unstructured meshes. The Lagrangian interpretation in this approach greatly reduces the time truncation errors in the Eulerian methods. To achieve high-order accuracy in the Galerkin-characteristics solver, the semi-Lagrangian method requires high-order interpolating procedures. In the present work, this step is carried out using the Bernstein-Bézier basis functions to evaluate the solution at the departure points. Triangular Bernstein-Bézier patches are constructed in a simple and inherent manner over finite elements along the characteristics. An efficient preconditioned conjugate gradient solver is used for the linear systems of algebraic equations. Several numerical examples including advection-diffusion equations with known analytical solutions and the viscous Burgers problem are considered to illustrate the accuracy, robustness and performance of the proposed approach. The computed results support our expectations for a stable and highly accurate Bernstein-Bézier Galerkin-characteristics finite element method for convection-diffusion problems.On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flowshttps://zbmath.org/1502.651182023-02-24T16:48:17.026759Z"García-Archilla, Bosco"https://zbmath.org/authors/?q=ai:garcia-archilla.bosco"John, Volker"https://zbmath.org/authors/?q=ai:john.volker"Novo, Julia"https://zbmath.org/authors/?q=ai:novo.juliaSummary: The kinetic energy of a flow is proportional to the square of the \(L^2(\Omega)\) norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree \(r\), then the best approximation error in \(L^2(\Omega)\) is of order \(r+1\). In this survey, the available finite element error analysis for the velocity error in \(L^\infty(0,T;L^2(\Omega))\) is reviewed, where \(T\) is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection-diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier-Stokes equations of order \(r-1,r\), and \(r+1/2\) for the velocity error in \(L^\infty(0,T;L^2(\Omega))\). All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf-sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open.Multiphysics finite element method for a nonlinear poroelasticity model with finite strainhttps://zbmath.org/1502.651192023-02-24T16:48:17.026759Z"Ge, Zhihao"https://zbmath.org/authors/?q=ai:ge.zhihao"Lou, Hui"https://zbmath.org/authors/?q=ai:lou.huiSummary: In this paper, we propose a fully discrete multiphysics finite element method to solve a nonlinear poroelasticity model with finite strain. To reveal the multi-physical processes of deformation and diffusion and propose a stable numerical method, we reformulate the original model into a fluid-fluid coupled problem -- a generalized nonlinear Stokes problem of displacement vector field and pseudo pressure field and a diffusion problem of other pseudo pressure field by a new technique. Then, we propose a multiphysics finite element method to approximate the spatial variables and use Newton method to solve the nonlinear problem, prove that the proposed numerical method is stable and has the optimal convergence orders, and give some numerical tests to show that the proposed numerical method is stable and has no oscillation for displacement and pressure. Finally, we draw conclusions to summary the main results of this work.Optimal control in a bounded domain for wave propagating in the whole space: coupling through boundary integral equationshttps://zbmath.org/1502.651202023-02-24T16:48:17.026759Z"Gong, Wei"https://zbmath.org/authors/?q=ai:gong.wei|gong.wei.1"Li, Buyang"https://zbmath.org/authors/?q=ai:li.buyang"Yang, Huanhuan"https://zbmath.org/authors/?q=ai:yang.huanhuanSummary: This paper is concerned with an optimal control problem in a bounded-domain \(\varOmega_0\) under the constraint of a wave equation in the whole space. The problem is regularized and then reformulated as an initial-boundary value problem of the wave equation in a bounded domain \(\varOmega \supset{\overline{\varOmega}}_0\) coupled with a set of boundary integral equations on \(\partial \varOmega\) taking account of wave propagation through the boundary. The well-posedness and stability of the reformulated problem are proved. A fully discrete finite element method is proposed for solving the reformulated problem. In particular, the wave equation in the bounded domain is discretized by an averaged central difference method in time, and the boundary integral equations are discretized in time by using the convolution quadrature generated by the second-order backward difference formula. The finite and boundary element methods are used for spatial discretization of the wave equation and the boundary integral equations, respectively. The stability and convergence of the numerical method are also proved. Finally, the spatial and temporal convergence rates are validated numerically in 2D.Well-balanced second-order convex limiting technique for solving the Serre-Green-Naghdi equationshttps://zbmath.org/1502.651212023-02-24T16:48:17.026759Z"Guermond, Jean-Luc"https://zbmath.org/authors/?q=ai:guermond.jean-luc"Kees, Chris"https://zbmath.org/authors/?q=ai:kees.chris"Popov, Bojan"https://zbmath.org/authors/?q=ai:popov.boyan"Tovar, Eric"https://zbmath.org/authors/?q=ai:tovar.eric-jSummary: In this paper, we introduce a numerical method for approximating the dispersive Serre-Green-Naghdi equations with topography using continuous finite elements. The method is an extension of the hyperbolic relaxation technique introduced in
[the authors, J. Comput. Phys. 450, Article ID 110809, 16 p. (2022; Zbl 07517103)]. It is explicit, second-order accurate in space, third-order accurate in time, and is invariant-domain preserving. It is also well balanced and parameter free. Special attention is given to the convex limiting technique when physical source terms are added in the equations. The method is verified with academic benchmarks and validated by comparison with laboratory experimental data.A robust and mass conservative virtual element method for linear three-field poroelasticityhttps://zbmath.org/1502.651222023-02-24T16:48:17.026759Z"Guo, Jun"https://zbmath.org/authors/?q=ai:guo.jun"Feng, Minfu"https://zbmath.org/authors/?q=ai:feng.minfuSummary: We present and analyze a robust and mass conservative virtual element method for the three-field formulation of Biot's consolidation problem in poroelasticity. The displacement and fluid flux are respectively approximated by enriched \(\boldsymbol{H}(\operatorname{div})\) virtual elements and \(\boldsymbol{H}(\operatorname{div})\) virtual elements, while the pressure is discretized by piecewise polynomial functions. Optimal a priori error estimates are obtained, including the semi-discrete scheme and the fully-discrete scheme with the implicit Euler approximation in time. Moreover, our method achieves robustness with respect to the constants hidden in the error estimates, even for the Lamé coefficient tending to infinity and the arbitrarily small constrained specific storage coefficient, and therefore it is free of both volumetric (Poisson) locking and nonphysical pressure oscillations. Meanwhile, it also conserves pointwise mass conservation for Biot's consolidation problem on the discrete level.Numerical solution of the heat transfer equation coupled with the Darcy flow using the finite element methodhttps://zbmath.org/1502.651232023-02-24T16:48:17.026759Z"Hirpho, Mohammed"https://zbmath.org/authors/?q=ai:hirpho.mohammedSummary: The finite element approach was utilized in this study to solve numerically the two-dimensional time-dependent heat transfer equation coupled with the Darcy flow. The Picard-Lindelöf Theorem was used to prove the existence and uniqueness of the solution. The prior and posterior error estimates are then derived for the numerical scheme. Numerical examples were provided to show the effectiveness of the theoretical results. The essential code development in this study was done using MATLAB computer simulation.Linear energy stable numerical schemes for a general chemo-repulsive modelhttps://zbmath.org/1502.651242023-02-24T16:48:17.026759Z"Jiang, Maosheng"https://zbmath.org/authors/?q=ai:jiang.maosheng"Zhao, Jia"https://zbmath.org/authors/?q=ai:zhao.jia"Wang, Qi"https://zbmath.org/authors/?q=ai:wang.qiSummary: We propose several novel numerical algorithms to solve a general chemo-repulsive model, which is used to study chemo-repulsive dynamics of biological species. We first introduce a new way to reformulate the chemo-repulsive model for chemotaxis into a gradient flow with a quadratic energy functional and a mobility matrix consisting of differential and integral operators. Secondly, we discretize the reformulated model using finite difference methods in time and finite element methods in space to obtain linear, second-order, energy stable numerical schemes. Then, we show that the schemes are unconditionally energy stable and the linear systems resulting from the schemes are uniquely solvable. Mesh refinement tests are conducted to confirm the convergence rates of the schemes and several numerical examples are given to demonstrate the usefulness of the schemes in simulating chemo-repulsive phenomena.Sparse Monte Carlo method for nonlocal diffusion problemshttps://zbmath.org/1502.651252023-02-24T16:48:17.026759Z"Kaliuzhnyi-Verbovetskyi, Dmitry"https://zbmath.org/authors/?q=ai:kaliuzhnyi-verbovetskyi.dmitry-s"Medvedev, Georgi S."https://zbmath.org/authors/?q=ai:medvedev.georgi-sA numerical method for solving nonlocal, nonlinear diffusion equations like
\[
u_t(t,x)=\int W(x,y)D(u(t,y)-u(t,x))dy+f(u,x,t)
\]
is proposed. The motivation for study such equations stems from modeling propagation phenomena in continuous media with nonlocal interactions. This numerical method combines sparse Monte Carlo and discontinuous Galerkin methods. Under suitable assumptions on the kernel \(W\), Lipschitz continuity of the function \(D\) and ``nonhomogeneous'' term \(f\), convergence of the proposed numerical method is shown together with estimations of errors. Numerical examples are provided.
Reviewer: Piotr Biler (Wrocław)A scalable exponential-DG approach for nonlinear conservation laws: with application to Burger and Euler equationshttps://zbmath.org/1502.651262023-02-24T16:48:17.026759Z"Kang, Shinhoo"https://zbmath.org/authors/?q=ai:kang.shinhoo"Bui-Thanh, Tan"https://zbmath.org/authors/?q=ai:bui-thanh.tanSummary: We propose an Exponential DG approach for numerically solving partial differential equations (PDEs). The idea is to decompose the governing PDE operators into linear (fast dynamics extracted by linearization) and nonlinear (the remaining after removing the former) parts, on which we apply the discontinuous Galerkin (DG) spatial discretization. The resulting semi-discrete system is then integrated using exponential time-integrators: exact for the former and approximate for the latter. By construction, our approach i) is stable with a large Courant number \((Cr>1)\); ii) supports high-order solutions both in time and space; iii) is computationally favorable compared to IMEX DG methods with no preconditioner; iv) requires comparable computational time compared to explicit RKDG methods, while having time stepsizes orders magnitude larger than maximal stable time stepsizes for explicit RKDG methods; v) is scalable in a modern massively parallel computing architecture by exploiting Krylov-subspace matrix-free exponential time integrators and compact communication stencil of DG methods. Various numerical results for both Burgers and Euler equations are presented to showcase these expected properties. For Burgers equation, we present a detailed stability and convergence analyses for the exponential Euler DG scheme.Galerkin finite element approximation of a stochastic semilinear fractional subdiffusion with fractionally integrated additive noisehttps://zbmath.org/1502.651272023-02-24T16:48:17.026759Z"Kang, Wenyan"https://zbmath.org/authors/?q=ai:kang.wenyan"Egwu, Bernard A."https://zbmath.org/authors/?q=ai:egwu.bernard-a"Yan, Yubin"https://zbmath.org/authors/?q=ai:yan.yubin"Pani, Amiya K."https://zbmath.org/authors/?q=ai:pani.amiya-kumarSummary: A Galerkin finite element method is applied to approximate the solution of a semilinear stochastic space and time fractional subdiffusion problem with the Caputo fractional derivative of the order \(\alpha \in (0, 1)\), driven by fractionally integrated additive noise. After discussing the existence, uniqueness and regularity results, we approximate the noise with the piecewise constant function in time, in order to obtain a regularized stochastic fractional subdiffusion problem. The regularized problem is then approximated by using the finite element method in spatial direction. The mean squared errors are proved based on the sharp estimates of the various Mittag-Leffler functions involved in the integrals. Numerical experiments are conducted to show that the numerical results are consistent with the theoretical findings.Error analysis for the pseudostress formulation of unsteady Stokes problemhttps://zbmath.org/1502.651282023-02-24T16:48:17.026759Z"Kim, Dongho"https://zbmath.org/authors/?q=ai:kim.dongho"Park, Eun-Jae"https://zbmath.org/authors/?q=ai:park.eun-jae"Seo, Boyoon"https://zbmath.org/authors/?q=ai:seo.boyoonSummary: In this paper, we are concerned with error analysis of the semi-discrete and fully discrete approximations to the pseudostress-velocity formulation of the unsteady Stokes problem. The pseudostress-velocity formulation of the Stokes problem allows a Raviart-Thomas mixed finite element. For the semi-discrete approximation, we prove that solution operators of homogeneous Stokes equations have the so-called parabolic smoothing property. For the fully discrete case, backward Euler and Crank-Nicolson schemes in time are considered. We present how to find the initial value of the pseudostress variable which is not given as initial data in Crank-Nicolson algorithm. Matrix equations are derived to show that backward Euler and Crank-Nicolson schemes corresponding to the pseudostress-velocity formulation are unconditionally stable. Finally, numerical examples are presented to test the performance of the algorithm and validity of the theory developed.A novel arbitrary Lagrangian-Eulerian finite element method for a mixed parabolic problem in a moving domainhttps://zbmath.org/1502.651292023-02-24T16:48:17.026759Z"Lan, Rihui"https://zbmath.org/authors/?q=ai:lan.rihui"Sun, Pengtao"https://zbmath.org/authors/?q=ai:sun.pengtaoSummary: In this paper, a novel arbitrary Lagrangian-Eulerian (ALE) mapping, thus a novel ALE-mixed finite element method (FEM), is developed and analyzed for a type of mixed parabolic equations in a moving domain. By means of a specific stabilization technique, the mixed finite element of a stable Stokes-pair is utilized to discretize this problem on the ALE description, and, stability and a nearly optimal convergence results are obtained for both semi- and fully discrete ALE finite element approximations. Numerical experiments are carried out to validate all theoretical results. The developed novel ALE-FEM can be also similarly extended to a transient porous (Darcy's) fluid flow problem in a moving domain as well as to Stokes/Darcy- or Stokes/Biot moving interface problem in the future.Conforming and nonconforming VEMs for the fourth-order reaction-subdiffusion equation: a unified frameworkhttps://zbmath.org/1502.651302023-02-24T16:48:17.026759Z"Li, Meng"https://zbmath.org/authors/?q=ai:li.meng"Zhao, Jikun"https://zbmath.org/authors/?q=ai:zhao.jikun"Huang, Chengming"https://zbmath.org/authors/?q=ai:huang.chengming"Chen, Shaochun"https://zbmath.org/authors/?q=ai:chen.shaochunSummary: We establish a unified framework to study the conforming and nonconforming virtual element methods (VEMs) for a class of time dependent fourth-order reaction-subdiffusion equations with the Caputo derivative. To resolve the intrinsic initial singularity we adopt the nonuniform Alikhanov formula in the temporal direction. In the spatial direction three types of VEMs, including conforming virtual element, \(C^0\) nonconforming virtual element and fully nonconforming Morley-type virtual element, are constructed and analysed. In order to obtain the desired convergence results, the classical Ritz projection operator for the conforming virtual element space and two types of new Ritz projection operators for the nonconforming virtual element spaces are defined, respectively, and the projection errors are proved to be optimal. In the unified framework we derive a prior error estimate with optimal convergence order for the constructed fully discrete schemes. To reduce the computational cost and storage requirements, the sum-of-exponentials (SOE) technique combined with conforming and nonconforming VEMs (SOE-VEMs) are built. Finally, we present some numerical experiments to confirm the theoretical correctness and the effectiveness of the discrete schemes. The results in this work are fundamental and can be extended into more relevant models.Local discontinuous Galerkin methods for diffusive-viscous wave equationshttps://zbmath.org/1502.651312023-02-24T16:48:17.026759Z"Ling, Dan"https://zbmath.org/authors/?q=ai:ling.dan"Shu, Chi-Wang"https://zbmath.org/authors/?q=ai:shu.chi-wang"Yan, Wenjing"https://zbmath.org/authors/?q=ai:yan.wenjingSummary: Numerical simulation of seismic wave equations has attracted much attention and plays a significant role in exploration seismology. As one of seismic wave models, the diffusive-viscous wave theory usually describes the attenuation of seismic wave propagating in fluid-saturated medium. In this paper, we focus on the design of numerical methods for the diffusive-viscous wave equations with variable coefficients. We develop a local discontinuous Galerkin (LDG) method, in which numerical fluxes are chosen carefully to maintain stability and accuracy. Moreover, we also prove the optimal error estimates for both the energy norm and the \(L^2\) norm. Numerical experiments are provided to demonstrate the optimal convergence rate and effectiveness of the proposed LDG method.Local and parallel finite element methods based on two-grid discretizations for a transient coupled Navier-Stokes/Darcy modelhttps://zbmath.org/1502.651322023-02-24T16:48:17.026759Z"Li, Qingtao"https://zbmath.org/authors/?q=ai:li.qingtao"Du, Guangzhi"https://zbmath.org/authors/?q=ai:du.guangzhiSummary: In this paper, some local and parallel finite element methods based on two-grid methods are presented for the non-stationary Navier-Stokes/Darcy model. Based on two-grid methods for spatial discretizations, both semi-discrete scheme and fully-discrete scheme with backward Euler method for the temporal discretization are proposed. Some local a priori estimate, which is crucial for the theoretical analysis, is obtained. The motivation of these local and parallel methods is that by utilizing decoupled method based on interface approximation via temporal extrapolation, low frequency could be obtained on the whole domain with a coarse grid, then solve some residual equations on some overlapped subdomains with a finer gird by some local and parallel procedures at each time step to catch high frequency. The interface coupling term on the subdomains with fine grid is approximated by the coarse-grid approximations on the previous time step. To overcome the global discontinuity of the numerical solution generated by the local and parallel finite element algorithms, a new parallel algorithm based on the partition of unity is developed. In the end, some numerical experiments are constructed to prove the effectiveness of our algorithms.Efficient fully decoupled and second-order time-accurate scheme for the Navier-Stokes coupled Cahn-Hilliard Ohta-Kawaski phase-field model of diblock copolymer melthttps://zbmath.org/1502.651332023-02-24T16:48:17.026759Z"Li, Tongmao"https://zbmath.org/authors/?q=ai:li.tongmao"Liu, Peng"https://zbmath.org/authors/?q=ai:liu.peng.1"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.11"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: In this work, we aim to develop a highly efficient numerical scheme for the flow-coupled phase-field model of diblock copolymer melt. Formally, the model is a very complicated nonlinear system that consists of the Navier-Stokes equations and the Cahn-Hilliard type equations with the Ohta-Kawaski potential. Through a combination of a novel decoupling technique and the projection method, we develop the first full decoupling, energy stable, and second-order time-accurate numerical scheme for this model. The decoupling technique is based on the design of an auxiliary ODE, which plays a vital role in obtaining the full decoupling structure while maintaining energy stability. The high efficiency of the scheme is not only reflected by its linear and decoupled structure but also because it only needs to solve a few elliptic equations at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Numerical experiments further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.Analysis of a Galerkin finite element method for the Maxwell-Schrödinger system under temporal gaugehttps://zbmath.org/1502.651342023-02-24T16:48:17.026759Z"Ma, Chupeng"https://zbmath.org/authors/?q=ai:ma.chupeng"Zhang, Yongwei"https://zbmath.org/authors/?q=ai:zhang.yongwei"Cao, Liqun"https://zbmath.org/authors/?q=ai:cao.liqunSummary: This paper is concerned with the numerical solution of the Maxwell-Schrödinger system under the temporal gauge, which describes light-matter interactions. We first propose a semidiscrete finite element scheme for the system and establish stability estimates for the finite element solution. Due to the lack of control over its divergence we cannot get \(\mathbf{H}^1\) \textit{a priori} estimates for the vector potential, making it difficult to obtain error estimates by usual techniques. We apply an exhaustion argument to overcome this difficulty and derive error estimates for the finite element approximation. An energy-conserving time-stepping scheme is proposed to solve the semidiscrete system.An iterative divergence-free immersed boundary method in the finite element framework for moving bodieshttps://zbmath.org/1502.651352023-02-24T16:48:17.026759Z"Mao, Jia"https://zbmath.org/authors/?q=ai:mao.jia.1"Zhao, Lanhao"https://zbmath.org/authors/?q=ai:zhao.lanhao"Liu, Xunnan"https://zbmath.org/authors/?q=ai:liu.xunnan"Mu, Kailong"https://zbmath.org/authors/?q=ai:mu.kailongSummary: A novel iterative direct-forcing immersed boundary (IB) method, which meets the divergence-free and no-slip conditions simultaneously, is proposed in the finite element framework for incompressible flow problems. The coupled velocity and the pressure of the fluid field are solved by the Characteristic-based Split scheme. The formulation for the extra body force is derived according to the no-slip boundary condition. The novelty of the proposed IB method lies in that the calculation of the pressure, the iterations of the velocity, the pressure and the extra body force are accomplished in a loop at the same time. Therefore, the divergence-free and no-slip conditions are ensured fully at the same time. The accuracy of the current IB method is verified by several cases, including static and moving boundaries. Meanwhile, the contributing factors of the spurious force oscillations for moving boundaries are studied. A brand-new viewpoint is proposed, which is different with the common standpoint. The major source of spurious force oscillations pointed out in this paper is that the change of interpolation coefficients when boundaries sweep within an element, instead of the change of interpolation points when boundaries move across adjacent elements, as discussed in existing references. The oscillations which are caused by the inherent properties of the immersed boundary method cannot be avoided, while effective methods can be employed to suppress them, including refining the meshes and applying a proper discrete delta function.Unconditional well-posedness and IMEX improvement of a family of predictor-corrector methods in micromagneticshttps://zbmath.org/1502.651362023-02-24T16:48:17.026759Z"Mauser, Norbert J."https://zbmath.org/authors/?q=ai:mauser.norbert-julius"Pfeiler, Carl-Martin"https://zbmath.org/authors/?q=ai:pfeiler.carl-martin"Praetorius, Dirk"https://zbmath.org/authors/?q=ai:praetorius.dirk"Ruggeri, Michele"https://zbmath.org/authors/?q=ai:ruggeri.micheleSummary: Recently, \textit{E. Kim} and \textit{J. Wilkening} [Q. Appl. Math. 76, No. 2, 383--405 (2018; Zbl 1387.78009)] proposed two novel predictor-corrector methods for the Landau-Lifshitz-Gilbert equation (LLG) in micromagnetics, which models the dynamics of the magnetization in ferromagnetic materials. Both integrators are based on the so-called Landau-Lifshitz form of LLG, use mass-lumped variational formulations discretized by first-order finite elements, and only require the solution of linear systems, despite the nonlinearity of LLG. The first(-order in time) method combines a linear update with an explicit projection of an intermediate approximation onto the unit sphere in order to fulfill the LLG-inherent unit-length constraint at the discrete level. In the second(-order in time) integrator, the projection step is replaced by a linear constraint-preserving variational formulation. In this paper, we extend the analysis of the integrators by proving unconditional well-posedness and by establishing a close connection of the methods with other approaches available in the literature. Moreover, the new analysis also provides a well-posed integrator for the Schrödinger map equation (which is the limit case of LLG for vanishing damping). Finally, we design an implicit-explicit strategy for the treatment of the lower-order field contributions, which significantly reduces the computational cost of the schemes, while preserving their theoretical properties.Fully-discrete spectral-Galerkin numerical scheme with second-order time accuracy and unconditional energy stability for the anisotropic Cahn-Hilliard modelhttps://zbmath.org/1502.651372023-02-24T16:48:17.026759Z"Min, Xilin"https://zbmath.org/authors/?q=ai:min.xilin"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.11"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: In this work, we construct a fully-discrete Spectral-Galerkin scheme for the anisotropic Cahn-Hilliard model. The scheme is based on the combination of a novel so-called explicit-Invariant Energy Quadratization method for time discretization and the Spectral-Galerkin approach for spatial discretization. The designed scheme only needs to solve several independent linear equations with constant coefficients at each time step, which demonstrates the high computational efficiency. The introduction of two auxiliary variables and the design of their associated auxiliary ODEs play a vital role in obtaining the linear structure and unconditional energy stability and thus avoiding the computation of a variable-coefficient system. The unconditional energy stability of the scheme is further rigorously proved, and the implementation process is given in detail. Through several 2D and 3D numerical simulations, we further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.Generalized multiscale hybridizable discontinuous Galerkin (GMsHDG) method for flows in nonlinear porous mediahttps://zbmath.org/1502.651382023-02-24T16:48:17.026759Z"Moon, Minam"https://zbmath.org/authors/?q=ai:moon.minamSummary: In this paper, we present a generalized multiscale hybridizable discontinuous Galerkin (GMsHDG) method for nonlinear porous media. We modified the spectral multiscale HDG framework introduced in Efendiev et al. (2015), to solve nonlinear problems. Also, we give projection-based error analysis on the GMsHDG method to numerically solve a nonlinear parabolic problem. The proposed method has two main ingredients: linearization and generating reduced dimensional multiscale spaces. The GMsHDG method yields that the error decreases when the eigenvalue of the local eigenvalue problem for generating a multiscale space increases, as demonstrated in the mathematical error analysis. Through representative numerical experiments, we confirm the reliability of the error estimations and show that the proposed method is practical and efficient.An unconditionally energy-stable linear Crank-Nicolson scheme for the Swift-Hohenberg equationhttps://zbmath.org/1502.651392023-02-24T16:48:17.026759Z"Qi, Longzhao"https://zbmath.org/authors/?q=ai:qi.longzhao"Hou, Yanren"https://zbmath.org/authors/?q=ai:hou.yanrenSummary: In this work, we propose and analyze a stabilized linear Crank-Nicolson scheme for the Swift-Hohenberg equation. More precisely, we treat the nonlinear term explicitly while two second-order stabilization terms are added to improve the stability of the scheme. We show that our scheme satisfies the discrete energy dissipation. We prove rigorously that our scheme is second-order accurate in time. Moreover, we adopt a spectral-Galerkin approximation for the spacial variables and establish error estimates for the fully discrete scheme. Numerical experiments are presented to show the accuracy and energy stability of our scheme.Numerical simulations of a dispersive model approximating free-surface Euler equationshttps://zbmath.org/1502.651402023-02-24T16:48:17.026759Z"Sánchez, Cipriano Escalante"https://zbmath.org/authors/?q=ai:sanchez.cipriano-escalante"Fernández-Nieto, Enrique D."https://zbmath.org/authors/?q=ai:fernandez-nieto.enrique-domingo"Morales de Luna, Tomás"https://zbmath.org/authors/?q=ai:morales-de-luna.tomas"Penel, Yohan"https://zbmath.org/authors/?q=ai:penel.yohan"Sainte-Marie, Jacques"https://zbmath.org/authors/?q=ai:sainte-marie.jacquesSummary: In some configurations, dispersion effects must be taken into account to improve the simulation of complex fluid flows. A family of free-surface dispersive models has been derived in [\textit{E. D. Fernández-Nieto} et al., Commun. Math. Sci. 16, No. 5, 1169--1202 (2018; Zbl 1408.35136)]. The hierarchy of models is based on a Galerkin approach and parameterised by the number of discrete layers along the vertical axis. In this paper we propose some numerical schemes designed for these models in a 1D open channel. The cornerstone of this family of models is the \textit{Serre-Green-Naghdi} model which has been extensively studied in the literature from both theoretical and numerical points of view. More precisely, the goal is to propose a numerical method for the \(LDNH_2\) model that is based on a projection method extended from the one-layer case to any number of layers. To do so, the one-layer case is addressed by means of a projection-correction method applied to a non-standard differential operator. A special attention is paid to boundary conditions. This case is extended to several layers thanks to an original relabelling of the unknowns. In the numerical tests we show the convergence of the method and its accuracy compared to the \(LDNH_0\) model.An online generalized multiscale finite element method for heat and mass transfer problem with artificial ground freezinghttps://zbmath.org/1502.651412023-02-24T16:48:17.026759Z"Spiridonov, Denis"https://zbmath.org/authors/?q=ai:spiridonov.denis"Stepanov, Sergei"https://zbmath.org/authors/?q=ai:stepanov.sergei-p"Vasiliy, Vasil'ev"https://zbmath.org/authors/?q=ai:vasiliy.vasilevSummary: The Online Generalized Multiscale Finite Element Method (Online GMsFEM) is presented in this study for heat and mass transfer problem in heterogeneous media with artificial ground freezing process. The mathematical model is based on the classical Stefan model, which depicts heat transfer with a phase change and includes filtration in a porous media. The model is described by a set of temperature and pressure equations. We employ a finite element method with the fictitious domain method to solve the problem on a fine grid. We apply a model reduction approach based on Online GMsFEM to derive a solution on the coarse grid. We can use the online version of GMsFEM to take less offline multiscale basis functions. We use decoupled offline basis functions built with snapshot space and based on spectral problems in our method. This is the standard approach of basis construction. We calculate additional basis functions in the offline stage to account for artificial ground freezing pipes. We use online multiscale basis functions to get a more precise approximation of phase change. We create an online basis that reduces error using local residual values. The accuracy of standard GMsFEM is greatly improved by using an online approach. Numerical results in a two-dimensional domain with layered heterogeneity are presented. To test the method's accuracy, we show results from a variety of offline and online basis functions. The results suggest that Online GMsFEM can deliver high-accuracy solutions with minimal processing resources.Application of C-Bézier and H-Bézier basis functions to numerical solution of convection-diffusion equationshttps://zbmath.org/1502.651422023-02-24T16:48:17.026759Z"Sun, Lanyin"https://zbmath.org/authors/?q=ai:sun.lanyin"Su, Fangming"https://zbmath.org/authors/?q=ai:su.fangmingSummary: Convection-diffusion equation is widely used to describe many engineering and physical problems. The finite element method is one of the most common tools for computing numerical solution. \textit{Q. Chen} and \textit{G. Wang} [Comput. Aided Geom. Des. 20, No. 1, 29--39 (2003; Zbl 1069.65514)] proposed C-Bézier and H-Bézier basis functions which are not only a generalization of classical Bernstein basis functions but also have a free shape parameter bringing a lot of flexibility to geometrical modeling. In this paper, we adopt C-Bézier and H-Bézier basis functions to construct test and trial function spaces of finite element method to get numerical solution of convection-diffusion equations. Compared with Lagrange basis functions, numerical accuracy is improved by 1--3 order-of magnitudes which implies a much better approximation in simulating convection-diffusion problems. Several examples are presented to verify the feasibility and effectiveness of our method.On energy laws and stability of Runge-Kutta methods for linear seminegative problemshttps://zbmath.org/1502.651432023-02-24T16:48:17.026759Z"Sun, Zheng"https://zbmath.org/authors/?q=ai:sun.zheng"Wei, Yuanzhe"https://zbmath.org/authors/?q=ai:wei.yuanzhe"Wu, Kailiang"https://zbmath.org/authors/?q=ai:wu.kailiangThe paper deals with continuous and discrete energy laws and stability analysis of explicit and implicit Runge-Kutta methods applied to autonomous linear seminegative differential systems. The authors generalize the theory developed in [\textit{Z. Sun} and \textit{C.-W. Shu}, SIAM J. Numer. Anal. 57, 175--192 (2019; Zbl 1422.65224)], where the strong stability was studied for explicit Runge-Kutta methods. First, continuous and discrete energy laws are derived for general Runge-Kutta methods. Since the derived discrete energy law characterizes the energy dissipation, it is used to propose criteria for strong and weak stability of Runge-Kutta methods. For several specific implicit Runge-Kutta methods, examples of discrete energy laws and stability analysis are given. Furthermore, the unified discrete energy law is derived for general diagonal Padé approximations of arbitrary order using the proposed theoretical framework and the Cholesky-type decomposition. For some special cases, the derived results are used to explain a connection between continuous and discrete energy law. Several numerical experiments confirm the validity of the derived theory.
Reviewer: Dana Černá (Liberec)Shifted Legendre reproducing kernel Galerkin method for the quasilinear degenerate parabolic problemhttps://zbmath.org/1502.651442023-02-24T16:48:17.026759Z"Su, Xuetong"https://zbmath.org/authors/?q=ai:su.xuetong"Yang, Jiabao"https://zbmath.org/authors/?q=ai:yang.jiabao"Yao, Huanmin"https://zbmath.org/authors/?q=ai:yao.huanminSummary: This paper focuses on solving a class of quasilinear degenerate parabolic problems by using shifted Legendre reproducing kernel Galerkin method (SLRKGM for short). The quasilinear term was linearized and the time derivative was discretized by using the finite difference scheme. The basis functions were constructed by the shifted Legendre polynomials and the approximate solution was obtained by the Galerkin method. We also discussed the error estimates and stability analysis of the method. Numerical examples demonstrate the feasibility and reliability of our algorithm.Pell collocation method for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernelhttps://zbmath.org/1502.651452023-02-24T16:48:17.026759Z"Taghipour, M."https://zbmath.org/authors/?q=ai:taghipour.mehran"Aminikhah, H."https://zbmath.org/authors/?q=ai:aminikhah.hossein(no abstract)A computational macroscale model for the time fractional poroelasticity problem in fractured and heterogeneous mediahttps://zbmath.org/1502.651462023-02-24T16:48:17.026759Z"Tyrylgin, Aleksei"https://zbmath.org/authors/?q=ai:tyrylgin.aleksei"Vasilyeva, Maria"https://zbmath.org/authors/?q=ai:vasilyeva.maria-v"Alikhanov, Anatoly"https://zbmath.org/authors/?q=ai:alikhanov.anatolii-alievich"Sheen, Dongwoo"https://zbmath.org/authors/?q=ai:sheen.dongwooSummary: In this paper, we consider the poroelasticity problem with a time memory formalism that couples the pressure and displacement, and we assume this multiphysics process occurs in multicontinuum media. A coupled system of equations for pressures in each continuum and elasticity equations for displacements of the medium are included in the mathematical model.
Based on the Caputo's time fractional derivative, we derive an implicit finite difference approximation for time discretization. Also, a Discrete Fracture Model (DFM) is used to model fluid flow through fractures and treat the complex network of fractures. Further, we develop a coarse grid approximation based on the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral problems for construction of the multiscale basis functions. The main idea of the proposed method is to reduce the dimensionality of the problem because our model equation has multiple fractional powers, there multiple unknowns with memory effects. Consequently, the solution is on a coarse grid, which saves some computational time.
We present numerical results for the two-dimensional model problems in fractured heterogeneous porous media. After, we investigate error analysis between reference (fine-scale) solution and multiscale solution with different numbers of multiscale basis functions. The results show that on a coarse grid, the proposed approach can achieve good accuracy.An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domainshttps://zbmath.org/1502.651472023-02-24T16:48:17.026759Z"von Wahl, Henry"https://zbmath.org/authors/?q=ai:von-wahl.henry"Richter, Thomas"https://zbmath.org/authors/?q=ai:richter.thomas-michael|richter.thomas"Lehrenfeld, Christoph"https://zbmath.org/authors/?q=ai:lehrenfeld.christophSummary: We analyse a Eulerian finite element method, combining a Eulerian time-stepping scheme applied to the time-dependent Stokes equations with the CutFEM approach using inf-sup stable Taylor-Hood elements for the spatial discretization. This is based on the method introduced by \textit{C. Lehrenfeld} and \textit{M. Olshanskii} [ESAIM, Math. Model. Numer. Anal. 53, No. 2, 585--614 (2019; Zbl 1422.65223)] in the context of a scalar convection-diffusion problems on moving domains, and extended to the nonstationary Stokes problem on moving domains by \textit{E. Burman} et al. [Numer. Math. 150, No. 2, 423--478 (2022; Zbl 07493698)] using stabilized equal-order elements. The analysis includes the geometrical error made by integrating over approximated level set domains in the discrete CutFEM setting. The method is implemented and the theoretical results are illustrated using numerical examples.Unconditional superconvergence analysis of an energy-stable finite element scheme for nonlinear Benjamin-Bona-Mahony-Burgers equationhttps://zbmath.org/1502.651482023-02-24T16:48:17.026759Z"Wang, Lele"https://zbmath.org/authors/?q=ai:wang.lele.1"Liao, Xin"https://zbmath.org/authors/?q=ai:liao.xin"Yang, Huaijun"https://zbmath.org/authors/?q=ai:yang.huaijunSummary: In this paper, an energy-stable Crank-Nicolson fully discrete finite element scheme is proposed for the Benjamin-Bona-Mahony-Burgers equation. Firstly, the stability of energy is proved, which leads to the boundedness of the finite element solution in \(H^1\)-norm. Secondly, combining with the above boundedness and the special property of bilinear element, the unconditional superclose and superconvergence results are derived. Finally, numerical examples are provided to illustrate the validity and efficiency of our theoretical analysis and method.Fully discrete spectral-Galerkin scheme for a ternary Allen-Cahn type mass-conserved Nakazawa-Ohta phase-field model for triblock copolymershttps://zbmath.org/1502.651492023-02-24T16:48:17.026759Z"Wang, Ziqiang"https://zbmath.org/authors/?q=ai:wang.ziqiang"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.11"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: In this paper, we first use the Allen-Cahn relaxation dynamics to simulate the phase-field model of triblock copolymers, where the mass-conserving property is achieved by adding two additional nonlocal Lagrange multipliers. The numerical approximations of the obtained model are further considered, and an efficient fully-discrete numerical scheme based on the Spectral-Galerkin approach for spatial discretization and the so-called explicit-IEQ (invariant energy quadratization) method for time marching is developed. Two auxiliary variables are introduced to reformulate the governing system into an equivalent form, which facilitates the design of numerical algorithms by simply discretizing nonlinear terms explicitly. The resulting scheme is not only second-order accurate in time and spectrally accurate in space, but also easy to implement, i.e., the scheme can be carried out by only solving multiple independently decoupled, linear, and constant-coefficient elliptic equations at each time step. We also prove the unconditional energy stability of the developed scheme and demonstrate its effectiveness by implementing several benchmarking numerical examples in 2D and 3D.An \(H^1\) weak Galerkin mixed finite element method for Sobolev equationhttps://zbmath.org/1502.651502023-02-24T16:48:17.026759Z"Xie, Chun-Mei"https://zbmath.org/authors/?q=ai:xie.chunmei"Feng, Min-Fu"https://zbmath.org/authors/?q=ai:feng.minfu"Wei, Hua-Yi"https://zbmath.org/authors/?q=ai:wei.huayiSummary: In this paper, we present a new \(H^1\) weak Galerkin mixed finite element method for the Sobolev equation which includes the exact solution \(u\) and the intermediate solution \(\mathbf{p} \). In the \(H^1\) weak Galerkin method, we adopt the discontinuous finite elements \(P_k / \mathbf{P}_k\) for the approximation solution pair \(( u_h , \mathbf{p}_h )\) on finite element partitions consisting of arbitrary shape of polygons which are WG shape regular. We give the semi-discrete and full-discrete formulations which are proven to be stable and parameter-free and possess the optimal error estimates. Numerical experiments show the efficiency of our methods.Optimal error estimates of Galerkin method for a nonlinear parabolic integro-differential equationhttps://zbmath.org/1502.651512023-02-24T16:48:17.026759Z"Yang, Huaijun"https://zbmath.org/authors/?q=ai:yang.huaijun"Shi, Dongyang"https://zbmath.org/authors/?q=ai:shi.dongyangSummary: In this paper, the error analysis of Galerkin finite element method (FEM) is investigated for a nonlinear parabolic integro-differential equation in two dimensions. By skillfully and rigorously manipulating the nonlinear term, optimal error estimates in \(L^\infty( L^2(\Omega))\) and \(L^\infty( H^1(\Omega))\) are obtained for a linearized backward Euler fully-discrete scheme, which improves the suboptimal approximation in \(L^\infty( L^2(\Omega))\) in the previous literature. Finally, some numerical results are provided to verify the theoretical findings.Numerical investigations of degenerate equations for fluid flow and reactive transport in clogging porous mediahttps://zbmath.org/1502.651522023-02-24T16:48:17.026759Z"Zech, Simon"https://zbmath.org/authors/?q=ai:zech.simon"Ray, Nadja"https://zbmath.org/authors/?q=ai:ray.nadja"Schulz, Raphael"https://zbmath.org/authors/?q=ai:schulz.raphaelSummary: Structural changes of the pore space and clogging phenomena are inherent to many porous media applications. However, numerical investigations of the related flow and transport problem remain challenging due to the degeneration of the hydrodynamic parameter porosity, and in turn of the permeability and diffusion tensor. A remedy is to use appropriate scalings of the unknowns velocity, pressure and species' concentration in terms of the porosity. This enables us to conduct numerical simulations for the combined, degenerating flow and transport problem. As discretization method, lowest order mixed finite elements are used and stability of the numerical scheme is shown. Under certain additional regularity assumptions, convergence (with optimal order) can be proven. Our numerical results confirm that optimal convergence is obtained for the transformed variables whereas the non-transformed variables might not converge.A moving finite element method for solving two-dimensional coupled Burgers' equations at high Reynolds numbershttps://zbmath.org/1502.651532023-02-24T16:48:17.026759Z"Zhang, Xiaohua"https://zbmath.org/authors/?q=ai:zhang.xiaohua"Xu, Xinmeng"https://zbmath.org/authors/?q=ai:xu.xinmengSummary: The coupled Burgers' equations at high Reynolds numbers usually have sharp gradients or are discontinuous in the solution. Therefore, it is difficult to obtain analytical solutions. This paper aims to use the moving finite element method proposed by \textit{R. Li} et al. [J. Comput. Phys. 170, No. 2, 562--588 (2001; Zbl 0986.65090)] to get stable and high-precision numerical solutions for the coupled Burgers' equations at high Reynolds numbers. The method decouples the mesh equation and partial differential equation (PDE) into two unrelated parts, mesh reconstruction and PDE solver. The mesh reconstruction constructs the harmonic mapping between the physical and logical domains through an iterative method so that the mesh structure maintains harmonics after multiple numerical integrations. We compute three classic numerical examples. Numerical results show that the moving finite element method effectively solve the coupled Burgers' equations at high Reynolds numbers, obtain stable numerical results, and achieve higher numerical accuracy. During the evolution of the solution, the mesh is always concentrated in the position where the solution has sharp gradients.A mixed virtual element method for the time-fractional fourth-order subdiffusion equationhttps://zbmath.org/1502.651542023-02-24T16:48:17.026759Z"Zhang, Yadong"https://zbmath.org/authors/?q=ai:zhang.yadong"Feng, Minfu"https://zbmath.org/authors/?q=ai:feng.minfuSummary: We propose and analyze a mixed virtual element method for time-fractional fourth-order subdiffusion equation involving the Caputo fractional derivative on polygonal meshes, whose solutions display a typical weak singularity at the initial time. By introducing an auxiliary variable \(\sigma = \Delta u\), then the fourth-order equation can be split into the coupled system of two second-order equations. Based on the \(L1\) scheme on a graded temporal mesh, the unconditional stability of the fully discrete is proved for two variables; and the priori error estimates are derived in \(L^2\) norm for the scalar unknown \(u\) and the variable \(\sigma \), respectively. Moreover, the priori error result in \(H^1\) semi-norm for the scalar unknown \(u\) also is obtained. Finally, a numerical calculation is implemented to verify the theoretical results.Local discontinuous Galerkin methods with generalized alternating numerical fluxes for two-dimensional linear Sobolev equationhttps://zbmath.org/1502.651552023-02-24T16:48:17.026759Z"Zhao, Di"https://zbmath.org/authors/?q=ai:zhao.di"Zhang, Qiang"https://zbmath.org/authors/?q=ai:zhang.qiang.2Summary: In this paper we present an efficient and high-order numerical method to solve two-dimensional linear Sobolev equations, which is based on the local discontinuous Galerkin (LDG) method with the upwind-biased fluxes and generalized alternating fluxes. A weak stability is given for both schemes, and a strong stability is established if the initial solutions exactly satisfy the elemental discontinuous Galerkin discretization. Moreover, the sharp error estimate in \(L^2\)-norm is established, by an elaborate application of the generalized Gauss-Radau projection. A fully-discrete LDG scheme is also considered, where the third-order explicit TVD Runge-Kutta algorithm is adopted. Finally some numerical experiments are given.Two-dimensional Haar wavelet method for numerical solution of delay partial differential equationshttps://zbmath.org/1502.651562023-02-24T16:48:17.026759Z"Amin, Rohul"https://zbmath.org/authors/?q=ai:amin.rohul"Patanarapeelert, Nichaphat"https://zbmath.org/authors/?q=ai:patanarapeelert.nichaphat"Barkat, Muhammad Awais"https://zbmath.org/authors/?q=ai:barkat.muhammad-awais"Mahariq, Ibrahim"https://zbmath.org/authors/?q=ai:mahariq.ibrahim"Sitthiwirattham, Thanin"https://zbmath.org/authors/?q=ai:sitthiwirattham.thanin(no abstract)An efficient kernel-based method for solving nonlinear generalized Benjamin-Bona-Mahony-Burgers equation in irregular domainshttps://zbmath.org/1502.651572023-02-24T16:48:17.026759Z"Azarnavid, Babak"https://zbmath.org/authors/?q=ai:azarnavid.babak"Emamjomeh, Mahdi"https://zbmath.org/authors/?q=ai:emamjomeh.mahdi"Nabati, Mohammad"https://zbmath.org/authors/?q=ai:nabati.mohammadSummary: An iterative kernel-based method is proposed for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers equation in regular and irregular domains. The method is based on the positive definite kernels pseudospectral method. We have used an iterative linearization scheme to overcome the nonlinearity of the problem. The convergence of the linearization scheme is proved. The space-time kernels are used for the full discretization of the problems in the temporal and spatial domains. Here, we construct the multidimensional kernels using the product of one-dimensional kernels. Numerical results and comparisons are presented for the several two and three-dimensional problems with the various domains. The obtained results confirm the efficiency and accuracy of the proposed method.Uniformly accurate splitting schemes for the Benjamin-Bona-Mahony equation with dispersive parameterhttps://zbmath.org/1502.651582023-02-24T16:48:17.026759Z"Calvo, Maria Cabrera"https://zbmath.org/authors/?q=ai:calvo.maria-cabrera"Schratz, Katharina"https://zbmath.org/authors/?q=ai:schratz.katharinaSummary: We propose a new class of uniformly accurate splitting methods for the Benjamin-Bona-Mahony equation which converge uniformly in the dispersive parameter \(\varepsilon \). The proposed splitting schemes are furthermore asymptotic convergent and preserve the KdV limit. We carry out a rigorous convergence analysis of the splitting schemes exploiting the smoothing properties in the system. This will allow us to establish improved error bounds with gain either in regularity (for non smooth solutions) or in the dispersive parameter \(\varepsilon \). The latter will be interesting in regimes of a small dispersive parameter. We will in particular show that in the classical BBM case \(P(\partial_x) = \partial_x\) our Lie splitting does not require any spatial regularity, i.e, first order time convergence holds in \(H^r\) for solutions in \(H^r\) without any loss of derivative. This estimate holds uniformly in \(\varepsilon \). In regularizing regimes \(\varepsilon =\mathscr{O}(1)\) we even gain a derivative with our time discretisation at the cost of loosing in terms of \(\frac{1}{\varepsilon}\). Numerical experiments underline our theoretical findings.A boundary meshless method for dynamic coupled thermoelasticity problemshttps://zbmath.org/1502.651592023-02-24T16:48:17.026759Z"Chen, Zhikang"https://zbmath.org/authors/?q=ai:chen.zhikang"Sun, Linlin"https://zbmath.org/authors/?q=ai:sun.linlin.1Summary: This work develops the boundary knot method (BKM) for two-dimensional (2D) coupled thermoelasticity problems in the frequency domain. By taking the non-singular general solution satisfying the governing equations as the basis function, the BKM does not require domain discretization. Nevertheless, the non-singular solution for the considered problems is absent. The Helmholtz decomposition and eigen-analysis are introduced to firstly derive the non-singular general solution, and thus the BKM can be formulated in terms of the discretized boundary points. In the final, the exponential window method (EWM) in conjunction with the BKM is employed to perform the transient analysis. The accuracy and feasibility of the proposed method are presented via numerical examples.Computational technique for heat and advection-diffusion equationshttps://zbmath.org/1502.651602023-02-24T16:48:17.026759Z"Jena, Saumya Ranjan"https://zbmath.org/authors/?q=ai:jena.saumya-ranjan"Gebremedhin, Guesh Simretab"https://zbmath.org/authors/?q=ai:gebremedhin.guesh-simretabSummary: In the literature, several techniques are implemented to obtain the approximate solution of heat and advection-diffusion equations. However, each method involves certain drawbacks such as high arithmetic computations, lower accuracy in terms of error, and difficult for computer programming. In the present work, the octic B-spline collocation approach is implemented to incorporate the drawback of the other numerical studies in the literature with with high accuracy in terms of error and MATLAB programming is executed to compute the tedious calculation in an easy way toward the improvement of the approximate solution of heat and advection-diffusion equation. The time derivative is discretized by forward difference technique and the Crank-Nicolson scheme is applied for the remaining terms of the advection-diffusion equation. The stability of the scheme is examined and found that the scheme is unconditionally stable. To test the accuracy and efficiency of the scheme, four test problems are computed. A better approximate solution is obtained as compared to existing methods and a good agreement on analytical solutions by the proposed scheme.An efficient hybrid numerical method for multi-term time fractional partial differential equations in fluid mechanics with convergence and error analysishttps://zbmath.org/1502.651612023-02-24T16:48:17.026759Z"Joujehi, A. Soltani"https://zbmath.org/authors/?q=ai:joujehi.a-soltani"Derakhshan, M. H."https://zbmath.org/authors/?q=ai:derakhshan.mohammad-hossein"Marasi, H. R."https://zbmath.org/authors/?q=ai:marasi.h-rSummary: The fundamental purpose of this paper is to study the numerical solution of multi-term time fractional nonlinear Klein-Gordon equation, using regularized beta functions and fractional order Bernoulli wavelets. First, the exact formulas for the fractional integrals of the fractional order Bernoulli wavelets were obtained. Using properties of the regularized beta functions and their operational matrices the operational matrices of the fractional order Bernoulli wavelets were calculated. Through new operational matrices and appropriate collocation points, the time fractional nonlinear Klein-Gordon equation were transformed to a system of nonlinear algebraic equations. The convergence analysis and error bound of the proposed method were then performed. A sufficient number of numerical simulations were considered to show the effectiveness and validity of the presented numerical method and its theoretical analysis.High-order conservative energy quadratization schemes for the Klein-Gordon-Schrödinger equationhttps://zbmath.org/1502.651622023-02-24T16:48:17.026759Z"Li, Xin"https://zbmath.org/authors/?q=ai:li.xin.18"Zhang, Luming"https://zbmath.org/authors/?q=ai:zhang.lumingSummary: In this paper, we design two classes of high-accuracy conservative numerical algorithms for the nonlinear Klein-Gordon-Schrödinger system in two dimensions. By introducing the energy quadratization technique, we first transform the original system into an equivalent one, where the energy is modified as a quadratic form. The Gauss-type Runge-Kutta method and the Fourier pseudo-spectral method are then employed to discretize the reformulation system in time and space, respectively. The fully discrete schemes inherit the conservation of mass and modified energy and can reach high-order accuracy in both temporal and spatial directions. In order to complement the proposed schemes and speed up the calculation, we also develop another class of conservative schemes combined with the prediction-correction technique. Numerous experimental results are reported to demonstrate the efficiency and high accuracy of the new methods.An efficient collocation algorithm with SSP-RK43 scheme to solve Rosenau-KdV-RLW equationhttps://zbmath.org/1502.651632023-02-24T16:48:17.026759Z"Shallu"https://zbmath.org/authors/?q=ai:shallu."Kukreja, Vijay Kumar"https://zbmath.org/authors/?q=ai:kukreja.vijay-kumarSummary: In this study, the coupling of the Rosenau-Korteweg-de Vries and Rosenau-regularized-long wave equations are solved that model the motion of shallow-water waves, by a collocation technique based on quintic B-spline as basis functions. For this, the spatial domain is discretized using the quintic B-spline collocation method, which leads to a system of first-order ordinary differential equations. A strong stability preserving Runge-Kutta method of four stages and third-order (SSP-RK43) is applied to solve the obtained system of equations. All the calculations are performed without any linearization or transformation. A couple of test problems are solved to show the efficacy and accuracy of the technique by calculating the \(L_2\) and \(L_{\infty}\) error norms as well as the discrete energy (\(\mathcal{E}\)) and mass (\(\mathcal{Q}\)) conservation properties. This equation is considered because it has great importance in the field of oceanography and the proposed technique is followed due to its ease of implementation and good accuracy. The stability analysis of the technique is performed using the concept of the Jacobian matrix along with eigenvalues and is shown to be stable. Comparison with the existing result shows that the proposed method is more accurate with a higher-order of convergence as compared to many existing techniques.Convergence analysis of Krylov subspace spectral methods for reaction-diffusion equationshttps://zbmath.org/1502.651642023-02-24T16:48:17.026759Z"Sheikholeslami, Somayyeh"https://zbmath.org/authors/?q=ai:sheikholeslami.somayyeh"Lambers, James V."https://zbmath.org/authors/?q=ai:lambers.james-v"Walker, Carley"https://zbmath.org/authors/?q=ai:walker.carleySummary: Krylov subspace spectral (KSS) methods are explicit time-stepping methods for partial differential equations that are designed to extend the advantages of Fourier spectral methods, when applied to constant-coefficient problems, to the variable-coefficient case. This paper presents a convergence analysis of a first-order KSS method applied to a system of coupled equations for modeling first-order photobleaching kinetics. The analysis confirms what has been observed in numerical experiments -- that the method is unconditionally stable and achieves spectral accuracy in space. Further analysis shows that this unconditional stability is not limited to the case in which the leading coefficient is constant.Solving anisotropic subdiffusion problems in annuli and shellshttps://zbmath.org/1502.651652023-02-24T16:48:17.026759Z"Tan, Jinying"https://zbmath.org/authors/?q=ai:tan.jinying"Liu, Jiangguo"https://zbmath.org/authors/?q=ai:liu.jiangguoSummary: This paper presents a family of numerical solvers for anisotropic subdiffusion problems in annuli and also cylindrical and spherical shells. The fractional order Caputo temporal derivative is discretized based on linear interpolation. The spatial Laplacian is discretized by utilizing Chebyshev and Fourier spectral collocation. Detailed discussion and useful formulas are presented for polar, cylindrical, and spherical coordinate systems. Numerical experiments along with a brief analysis are presented to demonstrate the accuracy and efficiency of these solvers. These solvers represent a continuation of our work in [\textit{J. Tan} and \textit{J. Liu}, J. Comput. Appl. Math. 364, Article ID 112318, 14 p. (2020; Zbl 07143626)] and shall be useful for numerical simulations of subdiffusion problems in cellular cytoplasm and other similar settings.Linearly implicit and second-order energy-preserving schemes for the modified Korteweg-de Vries equationhttps://zbmath.org/1502.651662023-02-24T16:48:17.026759Z"Yan, Jinliang"https://zbmath.org/authors/?q=ai:yan.jinliang"Zhu, Ling"https://zbmath.org/authors/?q=ai:zhu.ling"Lu, Fuqiang"https://zbmath.org/authors/?q=ai:lu.fuqiang"Zheng, Sihui"https://zbmath.org/authors/?q=ai:zheng.sihuiSummary: In this paper, some linearly implicit modified energy-conserving schemes are proposed for the modified Korteweg-de Vries equation (mKdV). The proposed schemes are based on the recently developed invariant energy quadratization (IEQ) approach and the scalar auxiliary variable (SAV) approach. We first introduce an auxiliary variable to transform the original model into an equivalent system, with a modified energy functional law. Then, the Fourier pseudospectral method is employed for the spatial discretization, and Crank-Nicolson, and Leap-Frog methods are used for the temporal discretization. We analyze the conservation properties, existence and uniqueness and the linear stability of the proposed schemes. The optimal order convergence rate of the semi-discrete scheme and the fully discrete schemes were analyzed, respectively. At last, some numerical examples are presented to illustrate the effectiveness of the proposed schemes.Randomized Newton's method for solving differential equations based on the neural network discretizationhttps://zbmath.org/1502.651672023-02-24T16:48:17.026759Z"Chen, Qipin"https://zbmath.org/authors/?q=ai:chen.qipin"Hao, Wenrui"https://zbmath.org/authors/?q=ai:hao.wenruiSummary: We develop a randomized Newton's method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton's method randomly chooses equations from the overdetermined nonlinear system resulting from the neural network discretization and solves the nonlinear system adaptively. We theoretically prove that the randomized Newton's method has a quadratic convergence locally. We also apply this new method to various numerical examples, from one to high-dimensional differential equations, to verify its feasibility and efficiency. Moreover, the randomized Newton's method can allow the neural network to ``learn'' multiple solutions for nonlinear systems of differential equations, such as pattern formation problems, and provides an alternative way to study the solution structure of nonlinear differential equations overall.Fast Huygens sweeping methods for a class of nonlocal Schrödinger equationshttps://zbmath.org/1502.651682023-02-24T16:48:17.026759Z"Ho, Kaho"https://zbmath.org/authors/?q=ai:ho.kaho"Leung, Shingyu"https://zbmath.org/authors/?q=ai:leung.shingyu"Qian, Jianliang"https://zbmath.org/authors/?q=ai:qian.jianliangSummary: We present efficient numerical methods for solving a class of nonlinear Schrödinger equations involving a nonlocal potential. Such a nonlocal potential is governed by Gaussian convolution of the intensity modeling nonlocal mutual interactions among particles. The method extends the Fast Huygens Sweeping Method (FHSM) that we developed in [\textit{S. Leung} et al., Methods Appl. Anal. 21, No. 1, 31--66 (2014; Zbl 1426.65151)] for the linear Schrödinger equation in the semi-classical regime to the nonlinear case with nonlocal potentials. To apply the methodology of FHSM effectively, we propose two schemes by using the Lie's and the Strang's operator splitting, respectively, so that one can handle the nonlinear nonlocal interaction term using the fast Fourier transform. The resulting algorithm can then enjoy the same computational complexity as in the linear case. Numerical examples demonstrate that the two operator splitting schemes achieve the expected first-order and second-order accuracy, respectively. We will also give one-, two- and three-dimensional examples to demonstrate the efficiency of the proposed algorithm.Comparison and analysis of natural laminar flow airfoil shape optimization results at transonic regime with bumps and trailing edge devices solved by Pareto games and EAshttps://zbmath.org/1502.651692023-02-24T16:48:17.026759Z"Chen, Yongbin"https://zbmath.org/authors/?q=ai:chen.yongbin"Tang, Zhili"https://zbmath.org/authors/?q=ai:tang.zhili"Periaux, Jacques"https://zbmath.org/authors/?q=ai:periaux.jacques-fSummary: The transonic natural laminar flow wing will become an important feature of the next generation advanced civil transport aircraft, because it can greatly reduce the friction drag. In paper [\textit{Z. Tang} et al., ``Solving the two objective evolutionary shape optimization of a natural laminar airfoil and shock control bump with game strategies'', Arch. Comput. Methods Eng. 26, No. 1, 119--141 (2019; \url{doi:10.1007/s11831-017-9231-6})] and [\textit{Y. B. Chen} et al., ``Trailing edge device application to wave drag reduction in NLF airfoil multi-island design optimization'', J. Nanjing Univ. Aeronaut. Astronaut. 50, No. 4, 548--557 (2018)], the problem of wave drag increase due to the expansion of laminar flow region in the optimization design of natural laminar airfoil is studied with Pareto game and EAs by using Shock Control Bump (SCB) and Trailing Edge Device (TED) respectively. In this paper, the numerical implementation of SCB and TED in the shape design optimization of natural laminar airfoils and the performance differences of the final optimal airfoil are compared and analyzed. The feasibility and potential of applying them to the optimization design of three-dimensional laminar wing are discussed.
For the entire collection see [Zbl 1471.65004].Error analysis for physics-informed neural networks (PINNs) approximating Kolmogorov PDEshttps://zbmath.org/1502.651702023-02-24T16:48:17.026759Z"De Ryck, Tim"https://zbmath.org/authors/?q=ai:de-ryck.tim"Mishra, Siddhartha"https://zbmath.org/authors/?q=ai:mishra.siddharthaSummary: Physics-informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely Kolmogorov equations that include the heat equation and Black-Scholes equation of option pricing, as examples. We construct neural networks, whose PINN residual (generalization error) can be made as small as desired. We also prove that the total \(L^2\)-error can be bounded by the generalization error, which in turn is bounded in terms of the training error, provided that a sufficient number of randomly chosen training (collocation) points is used. Moreover, we prove that the size of the PINNs and the number of training samples only grow polynomially with the underlying dimension, enabling PINNs to overcome the curse of dimensionality in this context. These results enable us to provide a comprehensive error analysis for PINNs in approximating Kolmogorov PDEs.Uniform error estimates for artificial neural network approximations for heat equationshttps://zbmath.org/1502.651712023-02-24T16:48:17.026759Z"Gonon, Lukas"https://zbmath.org/authors/?q=ai:gonon.lukas"Grohs, Philipp"https://zbmath.org/authors/?q=ai:grohs.philipp"Jentzen, Arnulf"https://zbmath.org/authors/?q=ai:jentzen.arnulf"Kofler, David"https://zbmath.org/authors/?q=ai:kofler.david"Šiška, David"https://zbmath.org/authors/?q=ai:siska.davidSummary: Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly rather high-dimensional partial differential equations (PDEs). Very recently, there have also been a number of rigorous mathematical results in the scientific literature, which examine the approximation capabilities of such deep learning-based approximation algorithms for PDEs. These mathematical results from the scientific literature prove in part that algorithms based on ANNs are capable of overcoming the curse of dimensionality in the numerical approximation of high-dimensional PDEs. In these mathematical results from the scientific literature, usually the error between the solution of the PDE and the approximating ANN is measured in the \(L^p\)-sense, with respect to some \(p \in [1, \infty)\) and some probability measure. In many applications it is, however, also important to control the error in a uniform \(L^\infty\)-sense. The key contribution of the main result of this article is to develop the techniques to obtain error estimates between solutions of PDEs and approximating ANNs in the uniform \(L^\infty\)-sense. In particular, we prove that the number of parameters of an ANN to uniformly approximate the classical solution of the heat equation in a region \([a, b]^d\) for a fixed time point \(T \in (0, \infty)\) grows at most polynomially in the dimension \(d\in\mathbb{N}\) and the reciprocal of the approximation precision \(\varepsilon > 0\). This verifies that ANNs can overcome the curse of dimensionality in the numerical approximation of the heat equation when the error is measured in the uniform \(L^\infty\)-norm.A nonlocal physics-informed deep learning framework using the peridynamic differential operatorhttps://zbmath.org/1502.651722023-02-24T16:48:17.026759Z"Haghighat, Ehsan"https://zbmath.org/authors/?q=ai:haghighat.ehsan"Bekar, Ali Can"https://zbmath.org/authors/?q=ai:bekar.ali-can"Madenci, Erdogan"https://zbmath.org/authors/?q=ai:madenci.erdogan"Juanes, Ruben"https://zbmath.org/authors/?q=ai:juanes.rubenSummary: The Physics-Informed Neural Network (PINN) framework introduced recently incorporates physics into deep learning, and offers a promising avenue for the solution of partial differential equations (PDEs) as well as identification of the equation parameters. The performance of existing PINN approaches, however, may degrade in the presence of sharp gradients, as a result of the inability of the network to capture the solution behavior globally. We posit that this shortcoming may be remedied by introducing long-range (nonlocal) interactions into the network's input, in addition to the short-range (local) space and time variables. Following this ansatz, here we develop a \textit{nonlocal} PINN approach using the Peridynamic Differential Operator (PDDO) -- a numerical method which incorporates long-range interactions and removes spatial derivatives in the governing equations. Because the PDDO functions can be readily incorporated in the neural network architecture, the nonlocality does not degrade the performance of modern deep-learning algorithms. We apply nonlocal PDDO-PINN to the solution and identification of material parameters in solid mechanics and, specifically, to elastoplastic deformation in a domain subjected to indentation by a rigid punch, for which the mixed displacement-traction boundary condition leads to localized deformation and sharp gradients in the solution. We document the superior behavior of nonlocal PINN with respect to local PINN in both solution accuracy and parameter inference, illustrating its potential for simulation and discovery of partial differential equations whose solution develops sharp gradients.A learning-based projection method for model order reduction of transport problemshttps://zbmath.org/1502.651732023-02-24T16:48:17.026759Z"Peng, Zhichao"https://zbmath.org/authors/?q=ai:peng.zhichao"Wang, Min"https://zbmath.org/authors/?q=ai:wang.min.4|wang.min|wang.min.1"Li, Fengyan"https://zbmath.org/authors/?q=ai:li.fengyanSummary: The Kolmogorov \(n\)-width of the solution manifolds of transport-dominated problems can decay slowly. As a result, it can be challenging to design efficient and accurate reduced order models (ROMs) for such problems. To address this issue, we propose a new learning-based projection method to construct nonlinear adaptive ROMs for transport problems. The construction follows the offline-online decomposition. In the offline stage, we train a neural network to construct adaptive reduced basis dependent on time and model parameters. In the online stage, we project the solution to the learned reduced manifold. Inheriting the merits from both deep learning and the projection method, the proposed method is more efficient than the conventional linear projection-based methods, and may reduce the generalization error of a solely learning-based ROM. Unlike some learning-based projection methods, the proposed method does not need to take derivatives of the neural network in the online stage.Adaptive integration of nonlinear evolution equations on tensor manifoldshttps://zbmath.org/1502.651742023-02-24T16:48:17.026759Z"Rodgers, Abram"https://zbmath.org/authors/?q=ai:rodgers.abram"Dektor, Alec"https://zbmath.org/authors/?q=ai:dektor.alec"Venturi, Daniele"https://zbmath.org/authors/?q=ai:venturi.danieleIn this paper, adaptive step-truncation algorithms are developed, selecting the tensor rank at each time step based on desired accuracy and stability constraints. These methods can be implemented easily as they rely only on arithmetic operations between tensors, which efficient and scalable parallel algorithms can perform. After defining a new criterion for tensor rank adaptivity based on local error estimates, the authors prove convergence of a wide range of rank-adaptive step-truncation algorithms. Furthermore, a connection between rank-adaptive step-truncation methods and rank-adaptive dynamical tensor approximation is established. Numerical results are presented and discussed for rank shock problem, two and four-dimensional Fokker-Planck equations using adaptive Euler, adaptive midpoint, and two-step rank-adaptive Adams-Bashforth method.
Reviewer: Bülent Karasözen (Ankara)Stochastic collocation method for stochastic optimal boundary control of the Navier-Stokes equationshttps://zbmath.org/1502.652292023-02-24T16:48:17.026759Z"Zhao, Wenju"https://zbmath.org/authors/?q=ai:zhao.wenju"Gunzburger, Max"https://zbmath.org/authors/?q=ai:gunzburger.max-dSummary: We consider the optimal control of a system governed by the Navier-Stokes equations with stochastic Dirichlet boundary conditions. Control conditions imposed only on the boundary are associated with reduced regularity of the system, as compared to distributed controls. To ensure the well-posedness of the solutions and the efficiency of numerical simulations, the stochastic boundary conditions and controls are required to belong almost surely to the Sobolev space of functions having first order weak derivative along the boundary. To simulate the system, numerical solutions are approximated using the stochastic collocation/finite element approach with sparse grid techniques and Monte Carlo methods which are applied to the boundary random field. An optimality system is derived for a matching-type cost functional. Error estimates are derived for the optimal state, the adjoint state and boundary control variables. Numerical examples for the deterministic cases are provided and compared in which the controls are applied on a part of or on the whole boundary. Simulations for the stochastic cases are also made with sparse grid and Monte Carlo methods to retrieve the statistical information of the optimal solution.Numerical approximation of the scattering amplitude in elasticityhttps://zbmath.org/1502.652302023-02-24T16:48:17.026759Z"Barceló, Juan A."https://zbmath.org/authors/?q=ai:barcelo.juan-antonio"Castro, Carlos"https://zbmath.org/authors/?q=ai:castro.carlos-a|castro.carlosSummary: We propose a numerical method to approximate the scattering amplitudes for the elasticity system with a non-constant matrix potential in dimensions \(d=2\) and 3. This requires to approximate first the scattering field, for some incident waves, which can be written as the solution of a suitable Lippmann-Schwinger equation. In this work we adapt the method introduced by \textit{G. Vainikko} [in: Direct and inverse problems of mathematical physics. Papers presented at special sessions of the ISAAC'97 congress, University of Delaware, Newark, DE, USA, June 2--7, 1997. Dordrecht: Kluwer Academic Publishers. 423--440 (2000; Zbl 0962.65097)] to solve such equations when considering the Lamé operator. Convergence is proved for sufficiently smooth potentials. Implementation details and numerical examples are also given.Size gradation control for anisotropic mixed-element mesh adaptationhttps://zbmath.org/1502.652332023-02-24T16:48:17.026759Z"Tenkes, Lucille-Marie"https://zbmath.org/authors/?q=ai:tenkes.lucille-marie"Alauzet, Frédéric"https://zbmath.org/authors/?q=ai:alauzet.fredericSummary: Metric-based mesh adaptation can be applied to hybrid mesh generation. Using a metric-orthogonal point-placement, a preliminary quasi-structured mesh is generated. Structured elements are then recovered in the most anisotropic areas. To this extent, it is necessary to ensure the smoothness of the metric field in the first place. This is achieved through a gradation correction process, that is the control of the size growth throughout the mesh. The smallest size prescriptions are spread using a metric intersection algorithm. In this paper, we demonstrate the relevance of size gradation control in our metric-based hybrid mesh generation process. Eventually, our goal is to design a gradation correction process that favors the alignment with the metric field, increases the number and improves the quality of the quadrilaterals. Several gradation control strategies are compared to determine which one is best-suited for hybrid mesh generation.A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographieshttps://zbmath.org/1502.760142023-02-24T16:48:17.026759Z"Bassi, C."https://zbmath.org/authors/?q=ai:bassi.caterina"Bonaventura, L."https://zbmath.org/authors/?q=ai:bonaventura.luca"Busto, S."https://zbmath.org/authors/?q=ai:busto.saray"Dumbser, M."https://zbmath.org/authors/?q=ai:dumbser.michaelSummary: We present a novel hyperbolic reformulation of the Serre-Green-Naghdi (SGN) model for the description of dispersive water waves. Contrarily to the classical Boussinesq-type models, it contains only \textit{first order} derivatives, thus allowing to overcome the numerical difficulties arising from higher order derivative terms, especially in the context of high order discontinuous Galerkin finite element schemes. The proposed model reduces to the original SGN model when an artificial sound speed tends to infinity. Moreover, it is endowed with an extra conservation law from which the energy-type conservation law associated with the original SGN model is retrieved when the artificial sound speed goes to infinity. In order to provide a theoretical basis for the proposed model, a derivation from the vertical average of the compressible Euler equations has been proposed. The governing partial differential equations are then solved at the aid of high order ADER discontinuous Galerkin finite element schemes. The new model has been successfully validated against numerical and experimental results, for both flat and non-flat bottom. For bottom topographies with large variations, the new model proposed in this paper provides more accurate results with respect to the hyperbolic reformulation of the SGN model with the mild bottom approximation recently proposed in [\textit{C. Escalante} et al., J. Comput. Phys. 394, 385--416 (2019; Zbl 1452.65188)].An adaptive well-balanced positivity preserving central-upwind scheme on quadtree grids for shallow water equationshttps://zbmath.org/1502.760152023-02-24T16:48:17.026759Z"Ghazizadeh, Mohammad A."https://zbmath.org/authors/?q=ai:ghazizadeh.mohammad-a"Mohammadian, Abdolmajid"https://zbmath.org/authors/?q=ai:mohammadian.abdolmajid-m"Kurganov, Alexander"https://zbmath.org/authors/?q=ai:kurganov.alexanderSummary: We present an adaptive well-balanced positivity preserving central-upwind scheme on quadtree grids for shallow water equations. The use of quadtree grids results in a robust, efficient and highly accurate numerical method. The quadtree model is developed based on the well-balanced positivity preserving central-upwind scheme proposed in [\textit{A. Kurganov} and \textit{G. Petrova}, Commun. Math. Sci. 5, No. 1, 133--160 (2007; Zbl 1226.76008)]. The designed scheme is well-balanced in the sense that it is capable of exactly preserving ``lake-at-rest'' steady states. In order to achieve this as well as to preserve positivity of water depth, a continuous piecewise bilinear interpolation of the bottom topography function is utilized. This makes the proposed scheme capable of modelling flows over discontinuous bottom topography. Local gradients are examined to determine new seeding points in grid refinement for the next timestep. Numerical examples demonstrate the promising performance of the central-upwind quadtree scheme.Simulation-driven optimization of high-order meshes in ALE hydrodynamicshttps://zbmath.org/1502.760582023-02-24T16:48:17.026759Z"Dobrev, Veselin"https://zbmath.org/authors/?q=ai:dobrev.veselin-a"Knupp, Patrick"https://zbmath.org/authors/?q=ai:knupp.patrick-m"Kolev, Tzanio"https://zbmath.org/authors/?q=ai:kolev.tzanio-v"Mittal, Ketan"https://zbmath.org/authors/?q=ai:mittal.ketan"Rieben, Robert"https://zbmath.org/authors/?q=ai:rieben.robert-n"Tomov, Vladimir"https://zbmath.org/authors/?q=ai:tomov.vladimir-zSummary: In this paper we propose tools for high-order mesh optimization and demonstrate their benefits in the context of multi-material Arbitrary Lagrangian-Eulerian (ALE) compressible shock hydrodynamic applications. The mesh optimization process is driven by information provided by the simulation which uses the optimized mesh, such as shock positions, material regions, known error estimates, etc. These simulation features are usually represented discretely, for instance, as finite element functions on the Lagrangian mesh. The discrete nature of the input is critical for the practical applicability of the algorithms we propose and distinguishes this work from approaches that strictly require analytical information. Our methods are based on node movement through a high-order extension of the Target-Matrix Optimization Paradigm (TMOP) of [\textit{P. Knupp}, ``Introducing the target-matrix paradigm for mesh optimization via node-movement'', Eng. Comput. 28, No. 4, 419--429 (2012; \url{doi:10.1007/s00366-011-0230-1})]. The proposed formulation is fully algebraic and relies only on local Jacobian matrices, so it is applicable to all types of mesh elements, in 2D and 3D, and any order of the mesh. We discuss the notions of constructing adaptive target matrices and obtaining their derivatives, reconstructing discrete data in intermediate meshes, node limiting that enables improvement of global mesh quality while preserving space-dependent local mesh features, and appropriate normalization of the objective function. The adaptivity methods are combined with automatic ALE triggers that can provide robustness of the mesh evolution and avoid excessive remap procedures. The benefits of the new high-order TMOP technology are illustrated on several simulations performed in the high-order ALE application BLAST [``BLAST: high-order finite element hydrodynamics'', LLNL code (2018), \url{https://computing.llnl.gov/projects/blast}].Moving least-squares aided finite element method (MLS-FEM): a powerful means to predict pressure discontinuities of multi-phase flow fields and reduce spurious currentshttps://zbmath.org/1502.760612023-02-24T16:48:17.026759Z"Mostafaiyan, Mehdi"https://zbmath.org/authors/?q=ai:mostafaiyan.mehdi"Wießner, Sven"https://zbmath.org/authors/?q=ai:wiessner.sven"Heinrich, Gert"https://zbmath.org/authors/?q=ai:heinrich.gertSummary: A technique based on the finite element method (FEM) is developed to precisely predict the pressure jump due to the surface tension forces in two-phase flow problems. In this method, the FEM shape functions of the active elements (the elements containing both phases) are enhanced by the aid of the moving least-squares (MLS) interpolation functions and used in the FEM calculations. Therefore, this technique is named as moving least-squares aided finite element method (MLS-FEM).
The enhanced shape function correlates the value of any unknown parameters (e.g., the velocity, pressure, or stress values) at any arbitrary point inside an active element to its surrounding nodes. When the enhancement is performed only for the pressure (P) shape functions, we briefly name the MLS-FEM as the PMLS method (pressure shape function enhanced by the MLS technique). In this case, it would be possible to predict the pressure discontinuities in a two-phase flow domain.
To assess the performance of the PMLS method in the calculation of the velocity components and pressure values in two-phase flow fields, the stationary drop problem is investigated. The results show that by employing the PMLS method, not only the magnitude of the spurious current is significantly reduced but also the pressure jump approaches toward its analytical value, without any pressure fluctuation at the interface. It is also explained that the method can be used to evaluate the extent of the drop deformation in a structured or an unstructured meshing system. Finally, the limitation of the introduced method is discussed, which is the basis for further developments.A high-order discontinuous Galerkin solver for the incompressible RANS equations coupled to the \(k\)-\(\varepsilon\) turbulence modelhttps://zbmath.org/1502.760632023-02-24T16:48:17.026759Z"Tiberga, Marco"https://zbmath.org/authors/?q=ai:tiberga.marco"Hennink, Aldo"https://zbmath.org/authors/?q=ai:hennink.aldo"Kloosterman, Jan Leen"https://zbmath.org/authors/?q=ai:kloosterman.jan-leen"Lathouwers, Danny"https://zbmath.org/authors/?q=ai:lathouwers.dannySummary: Accurate methods to solve the Reynolds-Averaged Navier-Stokes (RANS) equations coupled to turbulence models are still of great interest, as this is often the only computationally feasible approach to simulate complex turbulent flows in large engineering applications. In this work, we present a novel discontinuous Galerkin (DG) solver for the RANS equations coupled to the \(k - \epsilon\) model (in logarithmic form, to ensure positivity of the turbulence quantities). We investigate the possibility of modeling walls with a wall function approach in combination with DG. The solver features an algebraic pressure correction scheme to solve the coupled RANS system, implicit backward differentiation formulae for time discretization, and adopts the Symmetric Interior Penalty method and the Lax-Friedrichs flux to discretize diffusive and convective terms respectively. We pay special attention to the choice of polynomial order for any transported scalar quantity and show it has to be the same as the pressure order to avoid numerical instability. A manufactured solution is used to verify that the solution converges with the expected order of accuracy in space and time. We then simulate a stationary flow over a backward-facing step and a Von Kármán vortex street in the wake of a square cylinder to validate our approach.Application of projection and immersed boundary methods to simulating heat and mass transport in membrane distillationhttps://zbmath.org/1502.760652023-02-24T16:48:17.026759Z"Lou, Jincheng"https://zbmath.org/authors/?q=ai:lou.jincheng"Johnston, Jacob"https://zbmath.org/authors/?q=ai:johnston.jacob"Tilton, Nils"https://zbmath.org/authors/?q=ai:tilton.nilsSummary: Membrane distillation is an emerging desalination process with important applications to the energy-water nexus. Its performance depends, however, on heat and mass transport phenomena that are uniquely challenging to simulate. Difficulties include two adjacent channel flows coupled by heat and mass transport across a semi-permeable membrane. Within the channels, heat and mass boundary layers interact with the membrane surface and vortical flow structures generated by complicated geometries. The presence of multiple inlets and outlets also complicates the application of mass-conserving outlet conditions. Moreover, even small amounts of outlet noise affect the resolution of important near-membrane fluid velocities. We show these phenomena can be simulated to second-order spatial and temporal accuracy using finite volume methods with immersed boundaries and projection methods. Our approach includes a projection method that staggers the coupled channel flows and applies Robin boundary conditions to facilitate mass conservation at the outlets. We also develop an immersed boundary method that applies Neumann boundary conditions to second-order spatial accuracy. The methods are verified and validated against manufactured solutions and theoretical predictions of vortex shedding. They are then applied to the simulation of steady and unsteady transport phenomena in membrane distillation. The methods have important applications to the broad field of chemical engineering and deal with long-standing issues in both theoretical and computational fluid dynamics.Error analysis of a fully discrete consistent splitting MAC scheme for time dependent Stokes equationshttps://zbmath.org/1502.760662023-02-24T16:48:17.026759Z"Li, Xiaoli"https://zbmath.org/authors/?q=ai:li.xiaoli"Shen, Jie"https://zbmath.org/authors/?q=ai:shen.jie.4|shen.jie.3|shen.jie|shen.jie.5|shen.jie.2|shen.jie.1Summary: We construct and analyze a numerical scheme based on the truly consistent splitting approach in time and the MAC discretization in space for the time dependent Stokes equations. The scheme only requires solving several Poisson type equations for the velocity and pressure at each time step. We establish the equivalence between two different formulations of the fully discrete consistent splitting schemes, prove unconditional stability, and establish first-order in time and second-order in space error estimates for velocity and pressure in different discrete norms. We also provide numerical experiments to verify our theoretical results.Isogeometric semi-Lagrangian analysis for transport problemshttps://zbmath.org/1502.760782023-02-24T16:48:17.026759Z"Asmouh, Ilham"https://zbmath.org/authors/?q=ai:asmouh.ilham"El-Amrani, Mofdi"https://zbmath.org/authors/?q=ai:el-amrani.mofdi"Seaid, Mohammed"https://zbmath.org/authors/?q=ai:seaid.mohammedSummary: Isogeometric analysis (IGA) is combined with the semi-Lagrangian scheme to develop a stable and highly accurate method for the numerical solution of transport problems. An \(L^2\) projection using the non-uniform rational B-splines (NURBS) is proposed for the approximation of the solution at the departure points. The proposed method maintains the advantages of the semi-Lagrangian scheme in reducing the truncation errors and allowing for large CFL numbers in the simulations while the IGA guarantees the exact representation of the geometry of the computational domain. The performance of the isogeometric semi-Lagrangian analysis is demonstrated for a deformational flow problem and the benchmark of a single vortex flow.Optimized geometrical metrics satisfying free-stream preservationhttps://zbmath.org/1502.760792023-02-24T16:48:17.026759Z"Nolasco, Irving Reyna"https://zbmath.org/authors/?q=ai:nolasco.irving-reyna"Dalcin, Lisandro"https://zbmath.org/authors/?q=ai:dalcin.lisandro-d"Del Rey Fernández, David C."https://zbmath.org/authors/?q=ai:del-rey-fernandez.david-c"Zampini, Stefano"https://zbmath.org/authors/?q=ai:zampini.stefano"Parsani, Matteo"https://zbmath.org/authors/?q=ai:parsani.matteoSummary: Computational fluid dynamics and aerodynamics, which complement more expensive empirical approaches, are critical for developing aerospace vehicles. During the past three decades, computational aerodynamics capability has improved remarkably, following advances in computer hardware and algorithm development. However, for complex applications, the demands on computational fluid dynamics continue to increase in a quest to gain a few percent improvements in accuracy. Herein, we numerically demonstrate, in the context of tensor-product discretizations on hexahedral elements, that computing the metric terms with an optimization-based approach leads to a solution whose accuracy is overall on par and often better than the one obtained using the widely adopted \textit{P. D. Thomas} and \textit{C. K. Lombard} metric terms computation [AIAA J. 17, 1030--1037 (1979; Zbl 0436.76025)]. We show the efficacy of the proposed technique in the context of low and high-order accurate nonlinearly stable (entropy stable) schemes on distorted, high-order tensor product elements, considering smooth three-dimensional inviscid and viscous compressible test cases for which an analytical solution is known. The methodology, originally developed by \textit{J. Crean} et al. [J. Comput. Phys. 356, 410--438 (2018; Zbl 1380.76080)] in the context of triangular/tetrahedral grids, is not limited to tensor-product cells and it can be applied to other cell-based diagonal-norm summation-by-parts discretizations, including spectral differences, discontinuous Galerkin finite elements, and flux reconstruction schemes.Fast distance fields for fluid dynamics mesh generation on graphics hardwarehttps://zbmath.org/1502.760802023-02-24T16:48:17.026759Z"Roosing, Alo"https://zbmath.org/authors/?q=ai:roosing.alo"Strickson, Oliver"https://zbmath.org/authors/?q=ai:strickson.oliver"Nikiforakis, Nikos"https://zbmath.org/authors/?q=ai:nikiforakis.nikosSummary: We present a CUDA accelerated implementation of the Characteristic/Scan Conversion algorithm to generate narrow band signed distance fields in logically Cartesian grids. We outline an approach of task and data management on GPUs based on an input of a closed triangulated surface with the aim of reducing pre-processing and mesh-generation times. The work demonstrates a fast signed distance field generation of triangulated surfaces with tens of thousands to several million features in high resolution domains. We present improvements to the robustness of the original algorithm and an overview of handling geometric data.A posteriori error estimates for finite element discretizations of time-harmonic Maxwell's equations coupled with a non-local hydrodynamic drude modelhttps://zbmath.org/1502.780352023-02-24T16:48:17.026759Z"Chaumont-Frelet, T."https://zbmath.org/authors/?q=ai:chaumont-frelet.theophile"Lanteri, S."https://zbmath.org/authors/?q=ai:lanteri.stephane"Vega, P."https://zbmath.org/authors/?q=ai:vega.patrickSummary: We consider finite element discretizations of Maxwell's equations coupled with a non-local hydrodynamic Drude model that accurately accounts for electron motions in metallic nanostructures. Specifically, we focus on \textit{a posteriori} error estimation and mesh adaptivity, which is of particular interest since the electromagnetic field usually exhibits strongly localized features near the interface between metals and their surrounding media. We propose a novel residual-based error estimator that is shown to be reliable and efficient. We also present a set of numerical examples where the estimator drives a mesh adaptive process. These examples highlight the quality of the proposed estimator, and the potential computational savings offered by mesh adaptivity.Bright optical solitons with polynomial law of nonlinear refractive index by Adomian decomposition schemehttps://zbmath.org/1502.780372023-02-24T16:48:17.026759Z"González-Gaxiola, O."https://zbmath.org/authors/?q=ai:gonzalez-gaxiola.oswaldo"Biswas, Anjan"https://zbmath.org/authors/?q=ai:biswas.anjan"Yildirim, Yakup"https://zbmath.org/authors/?q=ai:yildirim.yakup"Alshehri, Hashim M."https://zbmath.org/authors/?q=ai:alshehri.hashim-mSummary: This paper numerically addresses bright optical solitons with cubic-quintic-septic (polynomial) law of nonlinear refractive index. The adopted scheme is with Adomian decomposition. The surface and contour plots are presented along with negligibly small error count.Effective mass theorems with Bloch modes crossingshttps://zbmath.org/1502.810322023-02-24T16:48:17.026759Z"Chabu, Victor"https://zbmath.org/authors/?q=ai:chabu.victor"Fermanian Kammerer, Clotilde"https://zbmath.org/authors/?q=ai:fermanian-kammerer.clotilde"Macià, Fabricio"https://zbmath.org/authors/?q=ai:macia.fabricioSummary: We study a Schrödinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wave-length comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make a rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands.Using incomplete Cholesky factorization to increase the time step in molecular dynamics simulationshttps://zbmath.org/1502.820252023-02-24T16:48:17.026759Z"Washio, Takumi"https://zbmath.org/authors/?q=ai:washio.takumi"Cui, Xiaoke"https://zbmath.org/authors/?q=ai:cui.xiaoke"Kanada, Ryo"https://zbmath.org/authors/?q=ai:kanada.ryo"Okada, Jun-ichi"https://zbmath.org/authors/?q=ai:okada.jun-ichi"Sugiura, Seiryo"https://zbmath.org/authors/?q=ai:sugiura.seiryo"Okuno, Yasushi"https://zbmath.org/authors/?q=ai:okuno.yasushi"Takada, Shoji"https://zbmath.org/authors/?q=ai:takada.shoji"Hisada, Toshiaki"https://zbmath.org/authors/?q=ai:hisada.toshiakiSummary: Incomplete Cholesky (IC) factorization is widely used as a preconditioner for accelerating the convergence of the conjugate gradient iterative method. The IC factorization has been used in continuum mechanics and other applications that require solutions of elliptic partial differential equations. In this study, we propose an efficient use of the IC factorization to increase the time step and accelerate molecular dynamics simulations. Previously, we proposed the semi-implicit Hessian correction (SimHec) scheme [the first author et al., ``Semi-implicit time integration with Hessian eigenvalue corrections for a larger time step in molecular dynamics simulations'', J. Chem. Theory Comput. 17, No. 9, 5792--5804 (2021; \url{doi:10.1021/acs.jctc.1c00398})] for overdamped Langevin dynamics of polymer simulations. SimHec constructs an approximation of the Hessian matrix by superposing the corrected elemental Hessian matrices associated with the interactions within a limited bandwidth along the polymer chain. The resulting narrow-bandwidth system is efficiently solved by an overlapped skyline solver in parallel. In this study, we integrate the non-local interactions in the polymer chain into the corrected Hessian matrix, and approximate the components of the Hessian matrix outside the narrow bandwidth using the IC factorization. This strategy allows us to use time steps that are 150--1000 times larger than those used in the explicit scheme without a severe increase in the computational load.On the stability of covariant BSSN formulationhttps://zbmath.org/1502.830022023-02-24T16:48:17.026759Z"Urakawa, Ryosuke"https://zbmath.org/authors/?q=ai:urakawa.ryosuke"Tsuchiya, Takuya"https://zbmath.org/authors/?q=ai:tsuchiya.takuya"Yoneda, Gen"https://zbmath.org/authors/?q=ai:yoneda.genSummary: In this study, we investigate the numerical stability of the covariant BSSN (cBSSN) formulation proposed by Brown. We calculate the constraint amplification factor (CAF), which is an eigenvalue of the coefficient matrix of the evolution equations of the constraints on the cBSSN formulation and on some adjusted formulations with constraints added to the evolution equations. The adjusted formulations have a higher numerical stability than the cBSSN formulation from the viewpoint of the CAF.A second order numerical method for the time-fractional Black-Scholes European option pricing modelhttps://zbmath.org/1502.910582023-02-24T16:48:17.026759Z"Kazmi, Kamran"https://zbmath.org/authors/?q=ai:kazmi.kamranSummary: In this paper, we design an efficient and accurate numerical method for solving the time-fractional Black-Scholes equation governing European options. The time-fractional Black-Scholes equation is transformed into an equivalent integro-differential equation. The numerical method for the integro-differential equation is developed by using a numerical integration scheme for time discretization and central difference formulas for space discretization. The stability and convergence of the method are analyzed. The numerical method is proved to be second-order accurate in both space and time. Richardson extrapolation is also introduced to obtain a modified version of the method that exhibits faster convergence for the time-fractional Black-Scholes equation with non-smooth initial data. Finally, two numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.