Recent zbMATH articles in MSC 65Mhttps://zbmath.org/atom/cc/65M2021-09-16T13:13:31.966056ZWerkzeugModel order reduction by proper orthogonal decompositionhttps://zbmath.org/1467.350212021-09-16T13:13:31.966056Z"Gräßle, Carmen"https://zbmath.org/authors/?q=ai:grassle.carmen"Hinze, Michael"https://zbmath.org/authors/?q=ai:hinze.michael"Volkwein, Stefan"https://zbmath.org/authors/?q=ai:volkwein.stefanSummary: We provide an introduction to proper orthogonal decomposition (POD) model order reduction with focus on (nonlinear) parametric partial differential equations (PDEs) and (nonlinear) time-dependent PDEs, and PDE-constrained optimization with POD surrogate models as application. We cover the relation of POD and singular value decomposition, POD from the infinite-dimensional perspective, reduction of nonlinearities, certification with a priori and a posteriori error estimates, spatial and temporal adaptivity, input dependency of the POD surrogate model, POD basis update strategies in optimal control with surrogate models, and sketch related algorithmic frameworks. The perspective of the method is demonstrated with several numerical examples.
For the entire collection see [Zbl 1455.93001].On a pore-scale stationary diffusion equation: scaling effects and correctors for the homogenization limithttps://zbmath.org/1467.350352021-09-16T13:13:31.966056Z"Khoa, Vo Anh"https://zbmath.org/authors/?q=ai:khoa.vo-anh"Thieu, Thi Kim Thoa"https://zbmath.org/authors/?q=ai:thieu.thi-kim-thoa"Ijioma, Ekeoma Rowland"https://zbmath.org/authors/?q=ai:ijioma.ekeoma-rowlandThe authors consider homogenization of the following semilinear elliptic equation in a periodically perforated domain \(\Omega_{\varepsilon}\) with a Fourier-type condition on the internal boundary \(\Gamma_{\varepsilon}\): \[-\nabla\cdot(A(x/\varepsilon)\nabla u_{\varepsilon})+\varepsilon^{\alpha}\mathcal{R}(u_{\varepsilon})=f(x) \mbox{ on }\Omega_{\varepsilon},\] \[-A(x/\varepsilon)\nabla u_{\varepsilon}\cdot n = \varepsilon^{\beta}\mathcal{S}(u_\epsilon) \mbox{ on }\Gamma_{\varepsilon},\] \[ u_{\varepsilon}=0\mbox{ on }\Gamma^{\text{ext}},\] where \(\mathcal{R}\) models a volume reaction and \(\mathcal{S}\) models a surface reaction and \(\Gamma^{\text{ext}}\) refers to the external boundary.
They prove well-posedness of the problem by a linearization scheme, under suitable hypothesis on the nonlinear terms \(\mathcal{R}\) and \(\mathcal{S}\).
Further, they construct asymptotic expansions in appropriate powers of \(\varepsilon\) for the solutions \(u_\varepsilon\) and prove higher order corrector estimates in three different regimes: (i) \(\alpha<0\) or \(\beta<0\), (ii) \(\alpha>0\) \& \(\beta>1\), and (iii) \(\alpha>0\) \& \(\beta\in[0,1)\). The construction of asymptotic expansions for \(u_{\varepsilon}\) requires the reaction terms \(\mathcal{R}\) and \(\mathcal{S}\) to have a matching asymptotic expansion. The theoretical results are supported by a numerical study.Well-posed variational formulations of Friedrichs-type systemshttps://zbmath.org/1467.351252021-09-16T13:13:31.966056Z"Berggren, Martin"https://zbmath.org/authors/?q=ai:berggren.martin"Hägg, Linus"https://zbmath.org/authors/?q=ai:hagg.linusSummary: All finite element methods, as well as much of the Hilbert-space theory for partial differential equations, rely on variational formulations, that is, problems of the type: find \(u \in V\) such that \(a(v, u) = l(v)\) for each \(v \in L\), where \(V, L\) are Sobolev spaces. However, for systems of Friedrichs type, there is a sharp disparity between established well-posedness theories, which are not variational, and the very successful discontinuous Galerkin methods that have been developed for such systems, which are variational. In an attempt to override this dichotomy, we present, through three specific examples of increasing complexity, well-posed variational formulations of boundary and initial-boundary-value problems of Friedrichs type. The variational forms we introduce are generalizations of those used for discontinuous Galerkin methods, in the sense that inhomogeneous boundary and initial conditions are enforced weakly through integrals in the variational forms. In the variational forms we introduce, the solution space is defined as a subspace \(V\) of the graph space associated with the differential operator in question, whereas the test function space \(L\) is a tuple of \(L^2\) spaces that separately enforce the equation, boundary conditions of characteristic type, and initial conditions.Correction to: ``An efficient linear scheme to approximate nonlinear diffusion problems''https://zbmath.org/1467.352022021-09-16T13:13:31.966056Z"Murakawa, Hideki"https://zbmath.org/authors/?q=ai:murakawa.hidekiFrom the text: The original article [the author, ibid. 35, No. 1, 71--101 (2018; Zbl 1400.35163)] has been published by the publisher without author's approval. The figures and citations in the publication of the original article were corrected.A conservative linearly-implicit compact difference scheme for the quantum Zakharov systemhttps://zbmath.org/1467.352702021-09-16T13:13:31.966056Z"Zhang, Gengen"https://zbmath.org/authors/?q=ai:zhang.gengen"Su, Chunmei"https://zbmath.org/authors/?q=ai:su.chunmeiSummary: This paper is devoted to developing and analysing a highly accurate conservative method for solving the quantum Zakharov system. The scheme is based on a linearly-implicit compact finite difference discretization and conserve the mass as well as energy in discrete level. Detailed numerical analysis is presented which shows the method is fourth-order accurate in space and second-order accurate in time. Several numerical examples are reported to confirm the conservation properties and high accuracy of the proposed scheme. Finally the compact scheme is applied to study the convergence rate of the quantum Zakharov system to its limiting model in the semi-classical limit.Vieta-Lucas polynomials for the coupled nonlinear variable-order fractional Ginzburg-Landau equationshttps://zbmath.org/1467.353052021-09-16T13:13:31.966056Z"Heydari, M. H."https://zbmath.org/authors/?q=ai:heydari.mohammad-hossien"Avazzadeh, Z."https://zbmath.org/authors/?q=ai:avazzadeh.zakieh"Razzaghi, M."https://zbmath.org/authors/?q=ai:razzaghi.mohsenSummary: In this article, the non-singular variable-order fractional derivative in the Heydari-Hosseininia concept is used to formulate the variable-order fractional form of the coupled nonlinear Ginzburg-Landau equations. To solve this system, a numerical scheme is constructed based upon the shifted Vieta-Lucas polynomials. In this method, with the help of classical and fractional derivative matrices of the shifted Vieta-Lucas polynomials (which are extracted in this study), solving the studied problem is transformed into solving a system of nonlinear algebraic equations. The convergence analysis and the truncation error of the shifted Vieta-Lucas polynomials in two dimensions are investigated. Numerical problems are demonstrated to confirm the convergence rate of the presented algorithm.Correction to: ``Stable and convergent fully discrete interior-exterior coupling of Maxwell's equations''https://zbmath.org/1467.353082021-09-16T13:13:31.966056Z"Nick, Jörg"https://zbmath.org/authors/?q=ai:nick.jorg"Kovács, Balázs"https://zbmath.org/authors/?q=ai:kovacs.balazs"Lubich, Christian"https://zbmath.org/authors/?q=ai:lubich.christianSummary: We correct a sign error in the paper [ibid. 137, No. 1, 91--117 (2017; Zbl 1375.35528)] by the second and third authors, noted by the first author. This sign error in the definition of the Calderón operator has no effect on the theory presented in [loc. cit.], but it does affect the implementation of the proposed numerical method.Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient-flow structurehttps://zbmath.org/1467.353172021-09-16T13:13:31.966056Z"Bailo, Rafael"https://zbmath.org/authors/?q=ai:bailo.rafael"Carrillo, José A."https://zbmath.org/authors/?q=ai:carrillo.jose-antonio"Hu, Jingwei"https://zbmath.org/authors/?q=ai:hu.jingweiSummary: We propose fully discrete, implicit-in-time finite-volume schemes for a general family of non-linear and non-local Fokker-Planck equations with a gradient-flow structure, usually known as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivity-preservation and energy-dissipation properties, essential for their practical use. The first-order scheme verifies these properties unconditionally for general non-linear diffusions and interaction potentials, while the second-order scheme does so provided a CFL condition holds. Sweeping dimensional splitting permits the efficient construction of these schemes in higher dimensions while preserving their structural properties. Numerical experiments validate the schemes and show their ability to handle complicated phenomena typical in aggregation-diffusion equations, such as free boundaries, metastability, merging and phase transitions.Wasserstein gradient flow formulation of the time-fractional Fokker-Planck equationhttps://zbmath.org/1467.353182021-09-16T13:13:31.966056Z"Duong, Manh Hong"https://zbmath.org/authors/?q=ai:duong.manh-hong"Jin, Bangti"https://zbmath.org/authors/?q=ai:jin.bangtiSummary: In this work, we investigate a variational formulation for a time-fractional Fokke-Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo derivative in time, and thus inherently nonlocal. The study follows the Wasserstein gradient flow approach pioneered by \textit{R. Jordan} et al. [SIAM J. Math. Anal. 29, No. 1, 1--17 (1998; Zbl 0915.35120)]. We propose a JKO-type scheme for discretizing the model, using the L1 scheme for the Caputo fractional derivative in time, and establish the convergence of the scheme as the time step size tends to zero. Illustrative numerical results in one- and two-dimensional problems are also presented to show the approach.Analysis of the linear sampling method for imaging penetrable obstacles in the time domainhttps://zbmath.org/1467.353432021-09-16T13:13:31.966056Z"Cakoni, Fioralba"https://zbmath.org/authors/?q=ai:cakoni.fioralba"Monk, Peter"https://zbmath.org/authors/?q=ai:monk.peter-b"Selgas, Virginia"https://zbmath.org/authors/?q=ai:selgas.virginiaSummary: We consider the problem of locating and reconstructing the geometry of a penetrable obstacle from time-domain measurements of causal waves. More precisely, we assume that we are given the scattered field due to point sources placed on a surface enclosing the obstacle and that the scattered field is measured on the same surface. From these multistatic scattering data we wish to determine the position and shape of the target.
To deal with this inverse problem, we propose and analyze the time-domain linear sampling method (TDLSM) by means of localizing the interior transmission eigenvalues in the Fourier-Laplace domain. We also prove new time-domain estimates for the forward problem and the interior transmission problem, as well as analyze several time-domain operators arising in the inversion scheme.Semiclassical limit of an inverse problem for the Schrödinger equationhttps://zbmath.org/1467.353442021-09-16T13:13:31.966056Z"Chen, Shi"https://zbmath.org/authors/?q=ai:chen.shi"Li, Qin"https://zbmath.org/authors/?q=ai:li.qinSummary: It is a classical derivation that the Wigner equation, derived from the Schrödinger equation that contains the quantum information, converges to the Liouville equation when the rescaled Planck constant \(\varepsilon\rightarrow 0\). Since the latter presents the Newton's second law, the process is typically termed the (semi-)classical limit. In this paper, we study the classical limit of an inverse problem for the Schrödinger equation. More specifically, we show that using the initial condition and final state of the Schrödinger equation to reconstruct the potential term, in the classical regime with \(\varepsilon\rightarrow 0\), becomes using the initial and final state to reconstruct the potential term in the Liouville equation. This formally bridges an inverse problem in quantum mechanics with an inverse problem in classical mechanics.Numerical solution of an axisymmetric inverse heat conduction problemhttps://zbmath.org/1467.353462021-09-16T13:13:31.966056Z"Djerrar, Ibtissem"https://zbmath.org/authors/?q=ai:djerrar.ibtissem"Alem, Leïla"https://zbmath.org/authors/?q=ai:alem.leila"Chorfi, Lahcène"https://zbmath.org/authors/?q=ai:chorfi.lahceneSummary: In this paper we are considering a boundary value axisymmetric inverse heat conduction problem inside the cylinder \(0 \leq r \leq b\). Our aim is to reconstruct the temperature \(f(t) =u(b,t)\) on the boundary \(r=b\) from measured temperature \(g^\delta (t)\) in the interior point \(0<r_1<b\). Using Laplace transform, for the direct problem the solution will be represented as a Fourier Bessel series. Then the inverse problem is reduced to an integral equation of type \(Af =g^\delta \). Such an equation is ill-posed, hence we use the Tikhonov regularization method. Finally, we conclude with numerical examples.
For the entire collection see [Zbl 1458.47004].An inverse problem for finding the lowest term of a heat equation with Wentzell-Neumann boundary conditionhttps://zbmath.org/1467.353492021-09-16T13:13:31.966056Z"Ismailov, Mansur I."https://zbmath.org/authors/?q=ai:ismailov.mansur-i"Tekin, Ibrahim"https://zbmath.org/authors/?q=ai:tekin.ibrahim"Erkovan, Sait"https://zbmath.org/authors/?q=ai:erkovan.saitThis paper deals with the heat equation \[ u_t =u_{xx} +p(t)u+f(x,t),\qquad (x,t)\in (0,1)\times(0,1), \] augmented with the initial condition \[ u(x,0)=\varphi(x),\qquad x\in(0,1), \] a Dirichlet boundary condition \[ u(0,t)=0,\qquad t\in(0,T), \] and a condition with non-local and/or non-classical type:
(1) a condition in which \(u(0, t)\) and/or \(u_x(0, t)\) is linearly related with \(u(1, t)\) and/or \(u_x(1, t)\) (non-local);
(2) a condition which contains the term of maximal order \(u_{xx}(0,t)\) or \(u_{xx}(1,t)\) (non- classical or Wentzell);
(3) a condition in which \(u(0, t)\) and/or \(u_x(0, t)\) is linearly related with \(u_{xx}(1, t)\) (non-local and non-classical).
Using the generalized Fourier method and under some regularity, consistency and orthogonality conditions on \(\varphi\) and \(f\), the authors prove the well-posedness of the classical solutions.Lattice Boltzmann method for stochastic convection-diffusion equationshttps://zbmath.org/1467.353682021-09-16T13:13:31.966056Z"Zhao, Weifeng"https://zbmath.org/authors/?q=ai:zhao.weifeng"Huang, Juntao"https://zbmath.org/authors/?q=ai:huang.juntao"Yong, Wen-An"https://zbmath.org/authors/?q=ai:yong.wen-anDiscrete maximal regularity for Volterra equations and nonlocal time-stepping schemeshttps://zbmath.org/1467.450032021-09-16T13:13:31.966056Z"Lizama, Carlos"https://zbmath.org/authors/?q=ai:lizama.carlos"Murillo-Arcila, Marina"https://zbmath.org/authors/?q=ai:murillo-arcila.marinaThe paper deals with a discrete-time formulation of a class of integro-partial differential equations with delay based on the following equation
\[
u(t)=\int_{-\infty}^{t}a(t-s)Au(s)ds+f(t,u(t)), \; t\in \mathbb{R}, \tag{1}
\]
where \(A\) is a closed linear operator defined on a Banach space \(X\).
The problem (1) can be formulated as the equivalent initial value problem
\[
u(t)=\int_{0}^{t}a(t-s)Au(s)ds+(Qu)(t)+f(t,u(t)),\;u(0)=\varphi(0), \tag{2}
\]
where \((Qu)(t)=\int_{-\infty}^{0}a(t-s)A\varphi(s)ds\).
The authors investigate two main questions concerning this problem:
\begin{itemize}
\item[(Q1)] Given a closed linear operator \(A\) and a kernel \(a\), for which time-discretization scheme of (2) there is maximal regularity in the \(l_p\)-setting?
\item[(Q2)] Which class of material-type functions are time-stepping schemes correlated with?
\end{itemize}
They introduce the new concept of 1-regular sequences that together with the \(\mathbb{R}\)-boundedness of the symbol of the equation ensures maximal regularity of the (linearized) discretization scheme (2) with zero-padding in the past time, in the \(l_p\)-setting. Using the new concept of 1-regularity, the earlier approaches to the first question are unified in a simpler way. A direct correlation between time-stepping schemes and material-type functions in viscoelasticity theory is also obtained for the second question.Adaptive virtual element method for optimal control problem governed by general elliptic equationhttps://zbmath.org/1467.490202021-09-16T13:13:31.966056Z"Wang, Qiming"https://zbmath.org/authors/?q=ai:wang.qiming"Zhou, Zhaojie"https://zbmath.org/authors/?q=ai:zhou.zhaojieSummary: In this paper a posteriori error analysis of virtual element method (VEM) for the optimal control problem governed by general elliptic equation is presented. The virtual element discrete scheme is constructed with virtual element approximation of the state equation and variational discretization of the control variable. Based on the a posteriori error estimates of virtual element method for general elliptic equation and approximated error equivalence of the solution of the optimal control problem to solutions of the state and adjoint problems we build up upper and lower a posteriori error estimates of the optimal control problem. Under the Dörfler's marking strategy, the traditional projected gradient algorithm and adaptive VEM algorithm drived by the state and adjoint error estimators are used to solve the optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equationhttps://zbmath.org/1467.600502021-09-16T13:13:31.966056Z"Prohl, Andreas"https://zbmath.org/authors/?q=ai:prohl.andreas"Wang, Yanqing"https://zbmath.org/authors/?q=ai:wang.yanqingSummaryDunst and Prohl, \textit{SIAM J. Sci. Comput.} 38 (2016) A2725-A2755: We verify strong rates of convergence for a time-implicit, finite-element based space-time discretization of the backward stochastic heat equation, and the forward-backward stochastic heat equation from stochastic optimal control. The fully discrete version of the forward-backward stochastic heat equation is then used within a gradient descent algorithm to approximately solve the linear-quadratic control problem for the stochastic heat equation driven by additive noise. This work is thus giving a theoretical foundation for the computational findings in [\textit{T. Dunst} and the first author, SIAM J. Sci. Comput. 38, No. 5, A2725--A2755 (2016; Zbl 1346.60106)].Stein variational reduced basis Bayesian inversionhttps://zbmath.org/1467.620422021-09-16T13:13:31.966056Z"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng"Ghattas, Omar"https://zbmath.org/authors/?q=ai:ghattas.omar-nA multifidelity ensemble Kalman filter with reduced order control variateshttps://zbmath.org/1467.621532021-09-16T13:13:31.966056Z"Popov, Andrey A."https://zbmath.org/authors/?q=ai:popov.andrey-a"Mou, Changhong"https://zbmath.org/authors/?q=ai:mou.changhong"Sandu, Adrian"https://zbmath.org/authors/?q=ai:sandu.adrian"Iliescu, Traian"https://zbmath.org/authors/?q=ai:iliescu.traianEntropy stable discontinuous Galerkin methods for nonlinear conservation laws on networks and multi-dimensional domainshttps://zbmath.org/1467.621872021-09-16T13:13:31.966056Z"Wu, Xinhui"https://zbmath.org/authors/?q=ai:wu.xinhui"Chan, Jesse"https://zbmath.org/authors/?q=ai:chan.jesseSummary: We present a high-order entropy stable discontinuous Galerkin method for nonlinear conservation laws on both multi-dimensional domains and on networks constructed from one-dimensional domains. These methods utilize treatments of multi-dimensional interfaces and network junctions which retain entropy stability when coupling together entropy stable discretizations. Numerical experiments verify the stability of the proposed schemes, and comparisons with fully 2D implementations demonstrate the accuracy of each type of junction treatment.Preconditioning for symmetric positive definite systems in balanced fractional diffusion equationshttps://zbmath.org/1467.650242021-09-16T13:13:31.966056Z"Fang, Zhi-Wei"https://zbmath.org/authors/?q=ai:fang.zhiwei"Lin, Xue-Lei"https://zbmath.org/authors/?q=ai:lin.xuelei"Ng, Michael K."https://zbmath.org/authors/?q=ai:ng.michael-k"Sun, Hai-Wei"https://zbmath.org/authors/?q=ai:sun.haiweiSummary: In this paper, we study the finite volume discretization method for balanced fractional diffusion equations where the fractional differential operators are comprised of both Riemann-Liouville and Caputo fractional derivatives. The main advantage of this approach is that new symmetric positive definite Toeplitz-like linear systems can be constructed for solving balanced fractional diffusion equations when diffusion functions are non-constant. It is different from non-symmetric Toeplitz-like linear systems usually obtained by existing numerical methods for fractional diffusion equations. The preconditioned conjugate gradient method with circulant and banded preconditioners can be applied to solve the proposed symmetric positive definite Toeplitz-like linear systems. Numerical examples, for both of one- and two- dimensional cases, are given to demonstrate the good accuracy of the finite volume discretization method and the fast convergence of the preconditioned conjugate gradient method.Hydrogen-helium chemical and nuclear galaxy collision: hydrodynamic simulations on AVX-512 supercomputershttps://zbmath.org/1467.650812021-09-16T13:13:31.966056Z"Chernykh, Igor"https://zbmath.org/authors/?q=ai:chernykh.igor"Kulikov, Igor"https://zbmath.org/authors/?q=ai:kulikov.igor-m"Tutukov, Alexander"https://zbmath.org/authors/?q=ai:tutukov.alexander-vThe authors study a computational hydrodynamic model of interacting galaxies. A model of gravitational multicomponent single-velocity hydrodynamics is employed to describe the interstellar medium. Further, a model based on the first moments of the collisionless Boltzmann equation describes the stellar component and dark matter. Subgrid processes of star formation and supernovae feedback, as well as cooling and heating functions are included in the model. The computational model takes into consideration, the chemical and nuclear reactions of the basic forms of hydrogen and helium before the formation of the helium hydride ion. The hydrodynamic equations have a suitable form for solving by a unified computational method based on the Harten-Lax-van Leer method. The governing equations and the computational scheme are written in vector form to use advanced vector extensions AVX-512. A parallel implementation of the model is presented. Results on the computational simulation experiments on the chemical dynamics of interacting galaxies are reported.A linearly implicit structure-preserving scheme for the fractional sine-Gordon equation based on the IEQ approachhttps://zbmath.org/1467.650822021-09-16T13:13:31.966056Z"Fu, Yayun"https://zbmath.org/authors/?q=ai:fu.yayun"Cai, Wenjun"https://zbmath.org/authors/?q=ai:cai.wenjun"Wang, Yushun"https://zbmath.org/authors/?q=ai:wang.yushunThe authors describe a numerical method for the (space-) fractional sine-Gordon equation in one space dimension. Adopting the invariant energy quadratization (IEQ) approach, they modify the equations and create a linearly-implicit method that conserves the modified energy discretely. They also present and analysis of the method as well as some numerical experiments.Correction to: ``An implicit-explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions''https://zbmath.org/1467.650832021-09-16T13:13:31.966056Z"Hochbruck, Marlis"https://zbmath.org/authors/?q=ai:hochbruck.marlis"Leibold, Jan"https://zbmath.org/authors/?q=ai:leibold.janCorrection to the authors' paper [ibid. 147, No. 4, 869--899 (2021; Zbl 1464.65086)].An artificial compressibility Crank-Nicolson leap-frog method for the Stokes-Darcy model and application in ensemble simulationshttps://zbmath.org/1467.650842021-09-16T13:13:31.966056Z"Jiang, Nan"https://zbmath.org/authors/?q=ai:jiang.nan"Li, Ying"https://zbmath.org/authors/?q=ai:li.ying.1|li.ying.3|li.ying.2|li.ying"Yang, Huanhuan"https://zbmath.org/authors/?q=ai:yang.huanhuanThis article discusses an unconditionally stable, second order convergent Crank-Nicolson scheme for numerically solving the Stokes-Darcy equations. The proposed approach decouples the Stokes-Darcy system into two smaller subphysics problems, which reduces the size of the linear systems to be solved at each time step, and allows parallel computing of the two detached problems. It is obtained that the method is unconditionally long time stable and second order convergent. This is also illustrated through a number of numerical experiments included in the article.A study of the numerical stability of an ImEx scheme with application to the Poisson-Nernst-Planck equationshttps://zbmath.org/1467.650852021-09-16T13:13:31.966056Z"Pugh, M. C."https://zbmath.org/authors/?q=ai:pugh.mary-c"Yan, David"https://zbmath.org/authors/?q=ai:yan.david"Dawson, F. P."https://zbmath.org/authors/?q=ai:dawson.f-pThe authors employ an adaptive time-stepper based on a second-order variable step-size, semi-implicit, backward differentiation formula (VSSBDF2) for solving the Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions (PNP-FBV). It is noted that, when the underlying dynamics is such that the solutions converge to a steady-state solution, the adaptive time-stepper produces solutions that ``nearly'' converge to the steady state and that, simultaneously, the time-step sizes stabilize to a limiting size. A linearization is considered and it is shown that this scheme is conditionally stable and that this is the cause of the adaptive time-stepper's behavior. Mesh-refinement, as well as a study of the eigenvectors corresponding to the critical eigenvalues, demonstrate that the conditional stability is not due to a time-step restriction caused by high-frequency contributions. The stability domain of the linearized scheme is considered and it is shown that its boundary can have corners as well as jump discontinuities.Sharp \(H^1\)-norm error estimates of two time-stepping schemes for reaction-subdiffusion problemshttps://zbmath.org/1467.650862021-09-16T13:13:31.966056Z"Ren, Jincheng"https://zbmath.org/authors/?q=ai:ren.jincheng"Liao, Hong-lin"https://zbmath.org/authors/?q=ai:liao.honglin"Zhang, Jiwei"https://zbmath.org/authors/?q=ai:zhang.jiwei"Zhang, Zhimin"https://zbmath.org/authors/?q=ai:zhang.zhiminThis article discusses a numerical approximation method for a reaction-diffusion equation involving fractional time derivative. The authors derive sharp \(H^1\)-error estimates by using a fractional Crank-Nicolson scheme combined with an improved Gronwall inequality. The numerical experiments included in the manuscript support the theorerical findings.Corrigendum to: ``A conservative linear difference scheme for the 2D regularized long-wave equation''https://zbmath.org/1467.650872021-09-16T13:13:31.966056Z"Wang, Xiaofeng"https://zbmath.org/authors/?q=ai:wang.xiaofeng.4"Dai, Weizhong"https://zbmath.org/authors/?q=ai:dai.weizhong"Guo, Shuangbing"https://zbmath.org/authors/?q=ai:guo.shuangbingSummary: The authors found a minor incorrect citation (Lemma 3.3) in their paper [ibid. 342, 55--70 (2019; Zbl 1429.65200)] which should be corrected in this Corrigendum.Large-time behaviour of a family of finite volume schemes for boundary-driven convection-diffusion equationshttps://zbmath.org/1467.650882021-09-16T13:13:31.966056Z"Chainais-Hillairet, Claire"https://zbmath.org/authors/?q=ai:chainais-hillairet.claire"Herda, Maxime"https://zbmath.org/authors/?q=ai:herda.maximeThe authors consider several convection-diffusion models set on a bounded domain with mixed nonhomogeneous Dirichlet and no-flux boundary conditions. These models include linear Fokker-Planck equations, nonlinear porous media equations and nonlinear systems of drift-diffusion equations. They satisfy a relative entropy principle at the continuous level, which allows to prove the exponential return to a steady state in the large time. The approximation of these equations is performed using finite volume methods and it is implicit in time. The numerical fluxes are discretized with two-point monotone fluxes written in the general framework of B-schemes, which allows to deal with upwind, centred and Schartfetter-Gummel discretizations. It is shown that the relative entropy principles also hold at the discrete level, and that the discrete solution to all these schemes exponentially decays to the discrete steady state of the scheme. This phenomna is illustrated numerically using the drift-diffusion system.A cell-centred pressure-correction scheme for the compressible Euler equationshttps://zbmath.org/1467.650892021-09-16T13:13:31.966056Z"Herbin, Raphaèle"https://zbmath.org/authors/?q=ai:herbin.raphaele"Latché, Jean-Claude"https://zbmath.org/authors/?q=ai:latche.jean-claude"Zaza, Chady"https://zbmath.org/authors/?q=ai:zaza.chadyThe authors consider a collocated finite volume method to approximate compressible Euler equations. A robust pressure-correction scheme is proposed. The scheme is based on an internal energy formulation, which ensures that the internal energy is positive. One of the main features of this scheme is that satisfies the fundamental stability properties. These stability properties yield in particular the existence of a solution to the scheme. It is proved a Lax-consistency-type convergence result, in the sense that, under some compactness assumptions, the limit of a converging sequence of approximate solutions obtained with space and time discretization steps tending to zero is an entropy weak solution of the Euler equations. Moreover, constant pressure and velocity are preserved through contact discontinuities. Some numerical tests are presented to support the finding theoretical results and the accuracy of the scheme. A numerical stabilization is introduced in order to reduce the oscillations which appear in some tests.Iterated fractional Tikhonov regularization method for solving the spherically symmetric backward time-fractional diffusion equationhttps://zbmath.org/1467.650902021-09-16T13:13:31.966056Z"Yang, Shuping"https://zbmath.org/authors/?q=ai:yang.shuping"Xiong, Xiangtuan"https://zbmath.org/authors/?q=ai:xiong.xiangtuan"Nie, Yan"https://zbmath.org/authors/?q=ai:nie.yanThe authors analyse an algorithm for recovering the distribution \(u(r,0)\) for an inhomogeneous time-fractional diffusion problem in a spherical symmetric domain \((0,R) \times (0,T)\), where the right-hand side of the differential equation and the function \(g\) in the condition \(u(r,T) = g(r)\), \(r \in [0,R]\) are noisy data. Hereby, the deterministic case and the random noise case are considered. An iterated version of the fractional Tikhonov regularization method is proposed. At first the deterministic situation is studied. Convergence estimates are proved in the case of an a priori choice of the regularization parameter and in the case of an a posteriori choice rule for the regularization parameter, where a modified discrepancy principle is used. In the random noise case a convergence estimate is given when an a priori choice rule is used for the regularization parameter. Finally, a numerical example is given to show the effectiveness of the proposed method.On superconvergence of Runge-Kutta convolution quadrature for the wave equationhttps://zbmath.org/1467.650912021-09-16T13:13:31.966056Z"Melenk, Jens Markus"https://zbmath.org/authors/?q=ai:melenk.jens-markus"Rieder, Alexander"https://zbmath.org/authors/?q=ai:rieder.alexanderAuthors' abstract: The semidiscretization of a sound soft scattering problem modelled by the wave equation is analyzed. The spatial treatment is done by integral equation methods. Two temporal discretizations based on Runge-Kutta convolution quadrature are compared: one relies on the incoming wave as input data and the other one is based on its temporal derivative. The convergence rate of the latter is shown to be higher than previously established in the literature. Numerical results indicate sharpness of the analysis. The proof hinges on a novel estimate on the Dirichlet-to-Impedance map for certain Helmholtz problems. Namely, the frequency dependence can be lowered by one power of \(|s|\) (up to a logarithmic term for polygonal domains) compared to the Dirichlet-to-Neumann map.Equilibrated flux \textit{a posteriori} error estimates in \(L^2(H^1)\)-norms for high-order discretizations of parabolic problemshttps://zbmath.org/1467.650922021-09-16T13:13:31.966056Z"Ern, Alexandre"https://zbmath.org/authors/?q=ai:ern.alexandre"Smears, Iain"https://zbmath.org/authors/?q=ai:smears.iain"Vohralík, Martin"https://zbmath.org/authors/?q=ai:vohralik.martinThe paper at hand provides a posteriori error estimates for the $L^2(H^1)$ error of approximations of the heat equation based on conforming hp-finite elements in space and discontinuous Galerkin schemes in time. The estimates are based on locally computable equilibrated flux reconstructions. The error estimates are guaranteed in the sense that they provide upper bounds for the $L^2(H^1)$ error without unknown constants. They are also locally efficient in the sense that the error estimator can be bounded from above by the $L^2(H^1)$ error plus temporal jumps under a rather mild condition on spatial and temporal mesh sizes, viz. $h^2 \leq \tau$.
The estimates employ an inf-sup theory for the continuous problem that provides equivalences of suitable norms of the error and of the residual.Numerical analysis of a parabolic hemivariational inequality for semipermeable mediahttps://zbmath.org/1467.650932021-09-16T13:13:31.966056Z"Han, Weimin"https://zbmath.org/authors/?q=ai:han.weimin"Wang, Cheng"https://zbmath.org/authors/?q=ai:wang.cheng.1This article discusses the numerical approximation of solutions to a class of parabolic hemivariational inequalities. The time derivative is approximated by the backward divided difference. Using the linear finite element method the authors obtain optimal error estimates for the approximation. Numerical experiments are included to support the theoretical findings.Space-time finite element discretization of parabolic optimal control problems with energy regularizationhttps://zbmath.org/1467.650942021-09-16T13:13:31.966056Z"Langer, Ulrich"https://zbmath.org/authors/?q=ai:langer.ulrich"Steinbach, Olaf"https://zbmath.org/authors/?q=ai:steinbach.olaf"Tröltzsch, Fredi"https://zbmath.org/authors/?q=ai:troltzsch.fredi"Yang, Huidong"https://zbmath.org/authors/?q=ai:yang.huidongIn this study, the authors have analyzed fully unstructured simplicial space-time meshes for the numerical solution of the parabolic optimal control problems with energy regularization in the Bochner space \(L^2(0,T;H^{-1}(\Omega))\). They have established the unique solvability in the continuous case with the help of Babuška's theorem. Further, they have proved the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Finally, they have given several numerical experiments to validate theoretical results.An oscillation-free discontinuous Galerkin method for scalar hyperbolic conservation lawshttps://zbmath.org/1467.650952021-09-16T13:13:31.966056Z"Lu, Jianfang"https://zbmath.org/authors/?q=ai:lu.jianfang"Liu, Yong"https://zbmath.org/authors/?q=ai:liu.yong.3|liu.yong.4|liu.yong.5|liu.yong.1|liu.yong.2"Shu, Chi-Wang"https://zbmath.org/authors/?q=ai:shu.chi-wangThe authors are concerned with a new strategy to control spurious oscillations in the DG methods when applied to scalar hyperbolic conservation laws. Actually, they introduce a damping term in the conventional DG scheme in order to control the high order terms. Their analysis shows that the proposed DG scheme maintains many good properties such as conservation, boundedness, and optimal error estimates. For one-dimensional linear problems, they additionally study the superconvergence behavior and for two-dimensional problems, they prove the optimal error estimates.
The authors carry out some numerical examples including both the linear and nonlinear scalar problems in one and two dimensions in order to underline the qualities of proposed method and mainly to show that it controls spurious numerical oscillations effectively.Discrete projections: a step towards particle methods on bounded domains without remeshinghttps://zbmath.org/1467.650962021-09-16T13:13:31.966056Z"Kirchhart, Matthias"https://zbmath.org/authors/?q=ai:kirchhart.matthias"Rieger, Christian"https://zbmath.org/authors/?q=ai:rieger.christianNumerical solutions of some stochastic hyperbolic wave equations including sine-Gordon equationhttps://zbmath.org/1467.650972021-09-16T13:13:31.966056Z"Tuckwell, Henry C."https://zbmath.org/authors/?q=ai:tuckwell.henry-cSummary: Many wave equations, including Klein-Gordon, Liouville's and the sine-Gordon equation, with added space-time white noise can be transformed to second order partial differential equations with mixed derivatives of the form \(Y_{xt} = F(Y) + \sigma W_{xt}\). Such equations are related to what \textit{G. J. Zimmerman} [Ann. Math. Stat. 43, 1235--1246 (1972; Zbl 0244.60032)] called a diffusion equation. For such equations an explicit numerical scheme is employed in both deterministic and stochastic examples which is checked for accuracy against known exact analytical deterministic solutions. The accuracy is further tested in the stochastic case with \(F=0\) by comparing statistics of solutions with those for the Brownian sheet. Generally the boundary conditions are chosen to be values of \(Y(x, 0)\) and \(Y(0, t)\) on boundaries of the first quadrant or subsets thereof, existence and uniqueness of solutions having been established for such systems. For the linear case, solutions are compared at various grid sizes and wave-like solutions were found, with and without noise, for non-zero initial and boundary conditions. Surprisingly, putative wave-like structures seemed to emerge with zero initial and boundary conditions and purely noise source terms. Equations considered with nonlinear \(F\) included quadratic and cubic together with the sine-Gordon equation. For the latter, wave-like structures were apparent with \(\sigma \leq 0.25\) but they tended to be shattered at larger values of \(\sigma\). Previous work on stochastic sine-Gordon equations is briefly reviewed.A Leray regularized ensemble-proper orthogonal decomposition method for parameterized convection-dominated flowshttps://zbmath.org/1467.650982021-09-16T13:13:31.966056Z"Gunzburger, Max"https://zbmath.org/authors/?q=ai:gunzburger.max-d"Iliescu, Traian"https://zbmath.org/authors/?q=ai:iliescu.traian"Schneier, Michael"https://zbmath.org/authors/?q=ai:schneier.michaelThe authors develop a Leray regularized ensemble-POD (proper orthogonal decomposition) method for the incompressible Navier-Stokes equations with perturbations in the forcing function, initial conditions and viscosities. The proposed algorithm is an ensemble-POD approach designed to work for moderate Reynolds number flows. The authors prove the stability and convergence of the finite element discretization of the Leray ensemble-POD model and by numerical simulation of two-dimensional flow past two offset cylinders, they show that the Leray ensemble-POD model is significantly more accurate than the standard ensemble-POD model.The computational complexity of elliptic partial differential equationshttps://zbmath.org/1467.651232021-09-16T13:13:31.966056Z"Schultz, Martin H."https://zbmath.org/authors/?q=ai:schultz.martin-hFrom the introduction: In this paper, we consider the computational complexity of the class of all procedures for computing a second order accurate approximation (on a square grid) to the solution of a linear, second order elliptic partial differential equation in a square domain. In particular, we present and analyze a new asymptotically optimal procedure for this problem. Moreover, we will show that a preconditioned form of our procedure is well-conditioned.
For the entire collection see [Zbl 1467.68010].Wave propagation in periodic buckled beams. I: Analytical models and numerical simulationshttps://zbmath.org/1467.740562021-09-16T13:13:31.966056Z"Maurin, Florian"https://zbmath.org/authors/?q=ai:maurin.florian-paul-robert"Spadoni, Alessandro"https://zbmath.org/authors/?q=ai:spadoni.alessandroSummary: Periodic buckled beams possess a geometrically nonlinear, load-deformation relationship and intrinsic length scales such that stable, nonlinear waves are possible. Modeling buckled beams as a chain of masses and nonlinear springs which account for transverse and coupling effects, homogenization of the discretized system leads to the Boussinesq equation. Since the sign of the dispersive and nonlinear terms depends on the level of buckling and support type (guided or pinned), compressive supersonic, rarefaction supersonic, compressive subsonic and rarefaction subsonic solitary waves are predicted, and their existence is validated using finite element simulations of the structure. Large dynamic deformations, which cannot be approximated with a polynomial of degree two, lead to strongly nonlinear equations for which closed-form solutions are proposed.Wave propagation in periodic buckled beams. Part II: Experimentshttps://zbmath.org/1467.740572021-09-16T13:13:31.966056Z"Maurin, Florian"https://zbmath.org/authors/?q=ai:maurin.florian-paul-robert"Spadoni, Alessandro"https://zbmath.org/authors/?q=ai:spadoni.alessandroSummary: Buckled periodic beams possess a geometrically nonlinear, load-deformation relation and intrinsic length scales such that stable, nonlinear waves are possible. In part I of this paper [Zbl 1467.74056], a model has been derived that predict compressive/rarefaction, supersonic/supersonic solitary waves, varying the level of compression and the support type (guided or pinned). Although this work has been validated by simulating the structure with finite-elements, in the present paper, investigations are done experimentally, focusing on the guided-supported, slightly-buckled beam.A rod-beam system with dynamic contact and thermal exchange conditionhttps://zbmath.org/1467.740642021-09-16T13:13:31.966056Z"Ahn, Jeongho"https://zbmath.org/authors/?q=ai:ahn.jeongho"Rail, Zachary"https://zbmath.org/authors/?q=ai:rail.zacharySummary: We study a mathematical hybrid dynamic system that models the setting, in which one end of a nonlinear viscoelastic Timoshenko beam is clamped while the another end is jointed with the bottom of a thermoviscoelastic rod. The top of the rod may come in contact with a rigid support. Two conditions are applied to the contacting end; Signorini's contact condition and Barber's heat exchange condition. We formulate a partial differential equation (PDE) system with the relevant boundary conditions that describes the motion of the combined rod-beam, taking into account the dynamic contact and thermal interaction. A variational formulation to the model is obtained in an abstract setting. Then, a hybrid of several numerical methods is employed to abstract formulations. It guarantees that all the conditions at each time step are satisfied. Convergence of numerical trajectories is shown, by passing to the limits as time step sizes approach zero. We derive a new energy balance form that plays an important role in establishing the numerical stability. The fully discrete numerical schemes are then used to compute numerical solutions and some representative numerical simulations are presented.Numerical resolution of an exact heat conduction model with a delay termhttps://zbmath.org/1467.740822021-09-16T13:13:31.966056Z"Campo, Marco"https://zbmath.org/authors/?q=ai:campo.marco-a"Fernandez, Jose R."https://zbmath.org/authors/?q=ai:fernandez.jose-ramon"Quintanilla, Ramon"https://zbmath.org/authors/?q=ai:quintanilla.ramonSummary: In this paper we analyze, from the numerical point of view, a dynamic thermoelastic problem. Here, the so-called exact heat conduction model with a delay term is used to obtain the heat evolution. Thus, the thermomechanical problem is written as a coupled system of partial differential equations, and its variational formulation leads to a system written in terms of the velocity and the temperature fields. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A priori error estimates are proved, from which the linear convergence of the algorithm could be derived under suitable additional regularity conditions. Finally, a two-dimensional numerical example is solved to show the accuracy of the approximation and the decay of the discrete energy.Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selectionhttps://zbmath.org/1467.740922021-09-16T13:13:31.966056Z"Liao, Hong-Lin"https://zbmath.org/authors/?q=ai:liao.honglin"Song, Xuehua"https://zbmath.org/authors/?q=ai:song.xuehua"Tang, Tao"https://zbmath.org/authors/?q=ai:tang.tao|tang.tao.1"Zhou, Tao"https://zbmath.org/authors/?q=ai:zhou.tao.3|zhou.tao.1|zhou.tao.2|zhou.taoSummary: In this work, we are concerned with the stability and convergence analysis of the second-order backward difference formula (BDF2) with variable steps for the molecular beam epitaxial model without slope selection. We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint. Then we show that it preserves an energy dissipation law if the adjacent time-step ratios satisfy \(r_k:=\tau_k/\tau_{k-1}<3.561\). Moreover, with a novel discrete orthogonal convolution kernels argument and some new estimates on the corresponding positive definite quadratic forms, the \(L^2\) norm stability and rigorous error estimates are established, under the same step-ratio constraint that ensures the energy stability, i.e., \(0<r_k<3.561\). This is known to be the best result in the literature. We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.A method for studying the dissipative properties of difference schemes in Euler coordinateshttps://zbmath.org/1467.760402021-09-16T13:13:31.966056Z"Shestakovskaya, Elena Sergeevna"https://zbmath.org/authors/?q=ai:shestakovskaya.elena-sergeevna"Starikov, Yaroslav Evgen'evich"https://zbmath.org/authors/?q=ai:starikov.yaroslav-evgenevich"Klinacheva, Nataliya Leonidovna"https://zbmath.org/authors/?q=ai:klinacheva.nataliya-leonidovnaSummary: At present, numerical methods for calculating shock-wave flows of liquid and gas in Eulerian coordinates have become widespread; therefore, the study of their characteristics is an urgent task. The paper presents an approach to assessing the dissipative properties of such difference schemes on strong discontinuities. The idea of the method is to construct the entropy production equation, the approximation error of which can be expressed by a combination of the approximation errors of the equations that make up the difference scheme. The equation of entropy production on a weak shock wave is used as a criterion for the dissipativity of the difference scheme. The paper evaluates the dissipative properties of the large particle method using the proposed method.No-slip and free-slip divergence-free wavelets for the simulation of incompressible viscous flowshttps://zbmath.org/1467.760412021-09-16T13:13:31.966056Z"Harouna, Souleymane Kadri"https://zbmath.org/authors/?q=ai:harouna.souleymane-kadri"Perrier, Valérie"https://zbmath.org/authors/?q=ai:perrier.valerieSummary: This work concerns divergence-free wavelet-based methods for the numerical resolution of Navier-Stokes equations. It generalizes to higher dimension the approach of \textit{S. Kadri Harouna} and \textit{V. Perrier} [Multiscale Model. Simul. 13, No. 1, 399--422 (2015; Zbl 1317.65266)] that reformulates the projection method using the Helmholtz-Hodge decomposition in wavelet domain. The solution is searched in a finite dimensional free-slip divergence-free wavelet space, with time-dependent wavelet coefficients. We prove and verify the convergence of a first-order time numerical scheme for the Helmholtz-Hodge-based projection method. Numerical simulations on the 3D lid-driven cavity flow show the accuracy and efficiency of the method.
For the entire collection see [Zbl 1459.65004].Single slit diffraction: from optics to elasticityhttps://zbmath.org/1467.780062021-09-16T13:13:31.966056Z"Berezovski, A."https://zbmath.org/authors/?q=ai:berezovski.arkadi"Engelbrecht, J."https://zbmath.org/authors/?q=ai:engelbrecht.juriSummary: The comparison of numerical simulations of the classical problem of the single slit diffraction in optical, acoustic, and elastic cases is presented in the paper in the plane strain setting. It is shown that wave fields downstream the slit are similar in optical and acoustic cases, as expected. Corresponding wave fields in the single slit diffraction using elastic materials become essentially different from optical and acoustic cases. This is an effect of elastic waves propagating inside the plate forming the slit.Convergence analysis of the PML method for time-domain electromagnetic scattering problemshttps://zbmath.org/1467.780162021-09-16T13:13:31.966056Z"Wei, Changkun"https://zbmath.org/authors/?q=ai:wei.changkun"Yang, Jiaqing"https://zbmath.org/authors/?q=ai:yang.jiaqing"Zhang, Bo"https://zbmath.org/authors/?q=ai:zhang.boThe perfectly matched layer (PML) method is proposed to solve the time-domain electromagnetic scattering problems in three dimensions. The PML problem is defined in a spherical layer and derived by using the Laplace transform and real coordinate stretching in the frequency domain. The well-posedness and the stability estimate of the PML problem are first proved based on the Laplace transform and the energy method. The exponential convergence of the PML method is then established in terms of the thickness of the layer and the PML absorbing parameter. The proof is mainly based on the stability estimates of solutions of the truncated PML problem and the exponential decay estimates of the stretched dyadic Green's function for the Maxwell equations in the free space.Numerical simulation of the formation of spherulites in polycrystalline binary mixtureshttps://zbmath.org/1467.820932021-09-16T13:13:31.966056Z"Hoppe, Ronald H. W."https://zbmath.org/authors/?q=ai:hoppe.ronald-h-w"Pahari, Basanta R."https://zbmath.org/authors/?q=ai:pahari.basanta-r"Winkle, James J."https://zbmath.org/authors/?q=ai:winkle.james-jSummary: Spherulites are growth patterns of average spherical form which may occur in the polycrystallization of binary mixtures due to misoriented angles at low grain boundaries. The dynamic growth of spherulites can be described by a phase field model where the underlying free energy depends on two phase field variables, namely the local degree of crystallinity and the orientation angle. For the solution of the phase field model we suggest a splitting scheme based on an implicit discretization in time which decouples the model and at each time step requires the successive solution of an evolutionary inclusion in the orientation angle and an evolutionary equation in the local degree of crystallinity. The discretization in space is done by piecewise linear Lagrangian finite elements. The fully discretized splitting scheme amounts to the solution of two systems of nonlinear algebraic equations. For the numerical solution we suggest a predictor-corrector continuation method with the discrete time as a parameter featuring constant continuation as a predictor and a semismooth Newton method for the first system and the classical Newton method for the second system as a corrector. This allows an adaptive choice of the time steps. Numerical results are given for the formation of a Category 1 spherulite.Thermodynamically consistent algorithms for models of incompressible multiphase polymer solutions with a variable mobilityhttps://zbmath.org/1467.821072021-09-16T13:13:31.966056Z"Shen, Xiaowen"https://zbmath.org/authors/?q=ai:shen.xiaowen"Wang, Qi"https://zbmath.org/authors/?q=ai:wang.qi.6|wang.qi|wang.qi.4|wang.qi.2|wang.qi.5|wang.qi.1|wang.qi.3Summary: We present a general strategy for developing structure and property preserving numerical algorithms for thermodynamically consistent models of incompressible multiphase polymer solutions with a variable mobility. We first present a formalism to derive thermodynamically consistent, incompressible, multiphase polymer models. Then, we develop the general strategy, known as the supplementary variable method, to devise thermodynamically consistent numerical approximations to the models. We illustrate the numerical strategy using newly developed models of incompressible diblock copolymer solutions coupled with an electric and a magnetic field, respectively. Mesh refinement is conducted to verify convergence rates of the developed schemes. Some numerical examples are given to exhibit underlying dynamics absent from and driven by the external fields, respectively, highlighting differences between models with the variable and constant mobilities.A pseudospectral method for option pricing with transaction costs under exponential utilityhttps://zbmath.org/1467.912112021-09-16T13:13:31.966056Z"de Frutos, Javier"https://zbmath.org/authors/?q=ai:de-frutos.javier"Gatón, Víctor"https://zbmath.org/authors/?q=ai:gaton.victorThe following dynamic model is known on the assumption that an investor holds an amount \(\overline X(t)\) in the bank account and \(\overline y(t)\) shares of a certain stock \(\overline S(t)\) \[ \begin{array}{l} d\overline X(t)=r \overline X(t)dt - (1+\lambda)\overline S(t)d\overline L(t)+(1-\mu)\overline S(t)d\overline M(t),\\
d\overline y(t)=d \overline L(t)-d\overline M(t),\\
d\overline S(t)=\overline S(t) \alpha dt+\overline S(t) \sigma dz_t,\\
\end{array} \] where \(r\) is the constant risk-free rate, \(\alpha\) is the constant expected rate of return of the stock, \(\sigma>0\) is the constant volatility of the stock and \(z_t\) is the standard Brownian motion. \(\overline L(t)\) and \(\overline M(t)\) are adapted, right-continuous, nonnegative and nondecreasing processes.
The authors propose two changes of variables that reduce the impact of the exponential growth and a Fourier pseudospectral method to solve the resulting nonlinear equation. Numerical analysis of the stability of the method is included.A PDE method for estimation of implied volatilityhttps://zbmath.org/1467.912132021-09-16T13:13:31.966056Z"Matić, Ivan"https://zbmath.org/authors/?q=ai:matic.ivan.1"Radoičić, Radoš"https://zbmath.org/authors/?q=ai:radoicic.rados"Stefanica, Dan"https://zbmath.org/authors/?q=ai:stefanica.danIn this article, the authors proved that the Black-Scholes integrated volatility \(\Sigma =\Sigma_{BS}(k,c)\) solves the following partial differential equations (PDE):
\[ \begin{array}{lcl} \Sigma ^3 \Sigma_{cc}=\Sigma_{c}^2\left (\frac{\Sigma ^4}{4}-k^2\right) \\
\Sigma^3 \Sigma_{kk}=\left (\frac{\Sigma^2 \Sigma_k}{2}+\Sigma-k\Sigma_k\right )\left (\frac{\Sigma^2 \Sigma_k}{2}-\Sigma+k\Sigma_k\right ) \\
\Sigma^3 \Sigma_{kc}=\Sigma_c\left (\frac{\Sigma^2}{2}-k\right )\left (\frac{\Sigma^2 \Sigma_k}{2}+k\Sigma_k-\Sigma\right ) \\
\end{array} \] where subscripts denote partial derivative \(\Sigma_c=\frac{\partial}{\partial c}\Sigma\) and \(\Sigma_k=\frac{\partial}{\partial k}\Sigma\).
The obtained PDE is then used to construct an algorithm for fast and accurate polynomial approximation for Black-Scholes implied volatility that improves on the existing numerical schemes from literature, both in speed and parallelizability.
An efficient and accurate numerical scheme for the calculation of implied Bachelier volatility is given in Section 9.Collocation method based on modified cubic B-spline for option pricing modelshttps://zbmath.org/1467.912142021-09-16T13:13:31.966056Z"Rashidinia, Jalil"https://zbmath.org/authors/?q=ai:rashidinia.jalil"Jamalzadeh, Sanaz"https://zbmath.org/authors/?q=ai:jamalzadeh.sanazThe authors develop an unconditional stable method based on modified cubic \(B\)-spline functions for the solution of option problems based on the Black-Scholes equation applied to European, American and barrier cases.
More precisely, a backward finite difference scheme is used for discretization of temporal derivative.
Then the modified \(B\)-spline approach is employed for approximating the option prices.
The stability of the constructed method has been proved in Section 3.
The efficiency of the proposed new method is tested by different examples.A compact finite difference scheme for fractional Black-Scholes option pricing modelhttps://zbmath.org/1467.912152021-09-16T13:13:31.966056Z"Roul, Pradip"https://zbmath.org/authors/?q=ai:roul.pradip"Goura, V. M. K. Prasad"https://zbmath.org/authors/?q=ai:prasad-goura.v-m-kIn this paper, the authors consider the following time-fractional Black-Scholes model \[ \begin{array}{l} \frac{\partial ^{\alpha} Y(S,\tau)}{\partial \tau ^{\alpha}}+0.5\sigma^2 S^2\frac{\partial ^2 Y(S,\tau)}{\partial S^2}+rS\frac{\partial Y(S,\tau)}{\partial S}-rY(S,\tau)=0,\\
[12pt] Y(0,\tau)=\tilde \phi (\tau),\: \: Y(\infty, \tau)=\tilde \omega(\tau)\\
\end{array} \] with terminal condition \(Y(S,T)=\psi (S)\), where \(Y\) denotes the price of a European option which depends on the asset price \(S\) and current time \(\tau\), \(\sigma\) is the volatility, \(r\) is the rick-free interest rate and \(\tau\) is the expiry time.
The time fractional derivative is described by means of Caputo and a compact finite difference method is employed for discretization of space derivative. Stability and convergence of the fully discrete scheme are studied in Section 4. The effect of fractional order derivative on the solution profile corresponding to option price is also analyzed.Where did the tumor start? An inverse solver with sparse localization for tumor growth modelshttps://zbmath.org/1467.920672021-09-16T13:13:31.966056Z"Subramanian, Shashank"https://zbmath.org/authors/?q=ai:subramanian.shashank"Scheufele, Klaudius"https://zbmath.org/authors/?q=ai:scheufele.klaudius"Mehl, Miriam"https://zbmath.org/authors/?q=ai:mehl.miriam"Biros, George"https://zbmath.org/authors/?q=ai:biros.georgeAnalysis and numerical simulations of a reaction-diffusion model with fixed active bodieshttps://zbmath.org/1467.920792021-09-16T13:13:31.966056Z"Yang, Chang"https://zbmath.org/authors/?q=ai:yang.chang"Tine, Léon Matar"https://zbmath.org/authors/?q=ai:tine.leon-matarState reconstruction of the wave equation with general viscosity and non-collocated observation and controlhttps://zbmath.org/1467.931552021-09-16T13:13:31.966056Z"Zheng, Fu"https://zbmath.org/authors/?q=ai:zheng.fu"Zhou, Hao"https://zbmath.org/authors/?q=ai:zhou.haoIt is considered the problem of reconstructing the initial condition (state) for the input-state-output \(1D\) system with distributed parameters described by hyperbolic partial differential equation and controlled/observed boundary conditions \[ \displaylines{\displaystyle{y_{tt} - y_{xx} + a(x)y_t = 0\ ;\ 0<x<1\;,\;t>0}\cr \displaystyle{y(0,t) = 0\ ,\ y_x(1,t) = u(t)}\cr O(t) = y_x(0,t)} \]
The problem is solved in terms of Luenberger observers, requiring analysis of stability and stabilization of the above system. Stabilization is achieved by local feedback \(u(t)=\alpha y_t(1,t)\) The paper also makes use of the approximation based on the method of lines.