Recent zbMATH articles in MSC 65N08https://zbmath.org/atom/cc/65N082022-10-04T19:40:27.024758ZWerkzeugFinite volume approximation of a two-phase two fluxes degenerate Cahn-Hilliard modelhttps://zbmath.org/1492.652502022-10-04T19:40:27.024758Z"Cancès, Clément"https://zbmath.org/authors/?q=ai:cances.clement"Nabet, Flore"https://zbmath.org/authors/?q=ai:nabet.floreThis article deals with the analysis of a time implicit finite volume scheme for the 2D degenerate Cahn-Hilliard model proposed in [\textit{W. E} and \textit{P. Palffy-Muhoray}, Phys. Rev. E (3) 55, No. 4, R3844--R3846 (1997; Zbl 1110.82304)]. This system describes the evolution of a mixture made of two incompressible phases, and its unknowns are \(c_i\) the volume fractions of the two phases and \(\mu_i\) the chemical potentials. It differs from the usual Cahn-Hilliard system by the addition of a nonlinear transport term driven by a divergence free vector field.
Since the proposed finite volume scheme is of TPFA (Two-Point Flux Approximation) type, an admissibility condition is required on the considered meshes. Moreover, an additional restriction with respect to the classical definition admissible meshes is enforced here (called super-admissibility), related to the construction of a strongly convergent SUSHI discrete gradient.
The proposed scheme is designed in such a way that the fundamental \textit{a priori} estimates satisfied at the continuous level can be transposed to the discrete setting. More precisely, the aforementioned properties are: mass conservation, positivity of the volume fractions, and energy decay. Moreover, since the established energy/energy dissipation estimate is not sufficient to obtain a control of the \(L^2\) norm of the chemical potentials, authors also need to quantify the production of mixing entropy to get this property.
Thanks to the established discrete \textit{a priori} estimates, authors are in position to prove their two main results: existence of a discrete solution (which is not straightforward since the scheme yields to a nonlinear system), and convergence towards a weak solution to the continuous problem. Existence proof relies on a topological degree argument, whereas the convergence is established thanks to compactness arguments, including in particular a discrete Aubin-Lions lemma detailed in Appendix B.
Finally, different numerical simulations are presented to illustrate the behavior of the introduced finite volume scheme.
Reviewer: Marianne Bessemoulin-Chatard (Nantes)Analysis of cell transmission model for traffic flow simulation with application to network traffichttps://zbmath.org/1492.652532022-10-04T19:40:27.024758Z"Maulana, A. S."https://zbmath.org/authors/?q=ai:maulana.a-s"Pudjaprasetya, S. R."https://zbmath.org/authors/?q=ai:pudjaprasetya.sri-redjekiSummary: The cell transmission model (CTM) is a macroscopic model that describes the dynamics of traffic flow over time and space. The effectiveness and accuracy of the CTM are discussed in this paper. First, the CTM formula is recognized as a finite-volume discretization of the kinematic traffic model with a trapezoidal flux function. To validate the constructed scheme, the simulation of shock waves and rarefaction waves as two important elements of traffic dynamics was performed. Adaptation of the CTM for intersecting and splitting cells is discussed. Its implementation on the road segment with traffic influx produces results that are consistent with the analytical solution of the kinematic model. Furthermore, a simulation on a simple road network shows the back and forth propagation of shock waves and rarefaction waves. Our numerical result agrees well with the existing result of Godunov's finite-volume scheme. In addition, from this accurately proven scheme, we can extract information for the average travel time on a certain route, which is the most important information a traveller needs. It appears from simulations of different scenarios that, depending on the circumstances, a longer route may have a shorter travel time. Finally, there is a discussion on the possible application for traffic management in Indonesia during the Eid al-Fitr exodus.An adaptive finite volume method for the diffraction grating problem with the truncated DtN boundary conditionhttps://zbmath.org/1492.652902022-10-04T19:40:27.024758Z"Wang, Zhoufeng"https://zbmath.org/authors/?q=ai:wang.zhoufengSummary: In this paper, we develop a adaptive finite volume method with the truncation of the nonlocal boundary operators for the wave scattering by periodic structures. The related truncation parameters are chosen through sharp a posteriori error estimate of the finite volume method. The crucial part of the a posteriori error analysis is to develop a duality argument technique and use a \(L^2\)-orthogonality property of the residual which plays a similar role as the Galerkin orthogonality. The a posteriori error estimate consists of two parts, the finite volume discretization error for adapting meshes and the truncation error of boundary operators which decays exponentially with respect to the truncation parameter \(N\). Numerical experiments are presented to confirm our theoretical analysis and show the efficiency and robustness of the proposed adaptive method.