European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   MS15: Novel flux approximation schemes for advection-diffusion problems (Part 1)
Chair: Martijn Anthonissen
08:30
25 mins
Robust discretization of advective linear transport, based on a complexte flux scheme and entropy principles
Alexander Linke, Jan ten Thije Boonkamp, Patricio Farrel
Abstract: A finite volume discretization for transient advective transport in one space dimension is presented, which merges a robust first-order upwind flux with a second-order complete flux. The (nonlinear) flux interpolation is based on entropy principles. Extensions of the scheme to two and three space dimensions are described
08:55
25 mins
Nonlinear free energy diminishing schemes for convection-diffusion equations: convergence and long time behaviour
Claire Chainais-Hillairet
Abstract: The large time behaviour of the solutions to linear convection-diffusion equations has been widely studied. It is known that some entropies are dissipated along time, so that, thanks to some functional inequalities, the exponential decay towards equilibrium is established for different kind of boundary conditions. When designing numerical schemes for these equations, it is crucial to ensure that the scheme has a similar large time behaviour than the continuous model. In \cite{CH}, Chainais-Hillairet and Herda prove that a family of TPFA finite volume schemes satisfies the exponential decay towards the associated discrete equilibrium. This family of B-schemes includes the classical centered, upwind and Scharfetter-Gummel schemes. Unfortunately, the B-schemes are based on two-point flux approximation and they can be used only on restricted meshes. In \cite{CCK}, Canc\`es, Chainais-Hillairet and Krell establish the convergence of a free-energy diminishing discrete duality finite volume (DDFV) scheme for the Fokker-Planck equations. Some numerical experiments show the exponential decay of the numerical scheme towards the thermal equilibrium. In this talk, we will introduce some nonlinear TPFA finite volume schemes for the Fokker-Planck equations. These schemes can be seen as the reduction of the nonlinear DDFV scheme introduced in \cite{CCK} to some admissible mesh. We will prove the exponential return to equilibrium of these schemes and give the main lines of the convergence proof. We will also explain how these results extend to the DDFV schemes from \cite{CCK}. \bibliographystyle{plain} \begin{thebibliography}{99} \bibitem{CCK} {\sc Cancès, C. ; Chainais-Hillairet, C. ; Krell, S.}, {\sl Numerical analysis of a nonlinear free-energy diminishing discrete duality finite volume scheme for convection diffusion equations.} Comput. Methods Appl. Math. 18 (2018), no. 3, 407--432. \bibitem{CH} {\sc Chainais-Hillairet, C. ; Herda, M.}, {\sl Large-time behavior of a family of finite volume schemes for boundary-driven convection-diffusion equations}, \url{https://hal.inria.fr/hal-01885015v1} \end{thebibliography}
09:20
25 mins
ENATE for complex domains I. Dirichlet boundary conditions
Antonio Pascau, Victor Javier Llorente
Abstract: ENATE (Enhanced Numerical Approximation of a Transport Equation) is a high-order scheme that provides the exact solution of the one-dimensional transport equation for arbitrary coefficients and source. In this paper how to deal with complex domains and Dirichlet boundary conditions is described under the framework of ENATE.
09:45
25 mins
A complete flux scheme based on multi-dimensional local boundary value problems
Martijn Anthonissen, Jan ten Thije Boonkkamp
Abstract: We consider the extension of one-dimensional complete flux schemes to more dimensions. In previous research this has been achieved by incorporating cross-fluxes into the source term and solving a quasi one-dimensional problem. This technique adds too much antidiffusion for three-dimensional problems. Another approach uses local orthogonal coordinates adapted to the flow, i.e., one coordinate axis is aligned with the local velocity field and the other one is perpendicular to it. This scheme shows good results but requires inter- and extrapolation of the solution. Here we base our flux approximation on exact solutions of local two-dimensional boundary problems allowing a better capturing of cross-flow and cross-diffusion than quasi one-dimensional methods.