A finite volume scheme for boundary-driven convection–diffusion equations with relative entropy structure

We propose a finite volume scheme for a class of nonlinear parabolic equations endowed with non-homogeneous Dirichlet boundary conditions and which admit relative entropy functionals. For this kind of models including porous media equations, Fokker–Planck equations for plasma physics or dumbbell models for polymer flows, it has been proved that the transient solution converges to a steady-state when time goes to infinity. The present scheme is built from a discretization of the steady equation and preserves steady-states and natural Lyapunov functionals which provide a satisfying long-time behavior. After proving well-posedness, stability, exponential return to equilibrium and convergence, we present several numerical results which confirm the accuracy and underline the efficiency to preserve large-time asymptotic.

[1]  R. Eymard,et al.  Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces , 2008, 0801.1430.

[2]  A. Arnold,et al.  On generalized Csiszár-Kullback inequalities , 2000 .

[3]  Qiangchang Ju,et al.  Large-time behavior of non-symmetric Fokker-Planck type equations , 2008 .

[4]  Mark A. Peletier,et al.  Variational modelling : energies, gradient flows, and large deviations , 2014, 1402.1990.

[5]  Marianne Bessemoulin-Chatard,et al.  A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme , 2010, Numerische Mathematik.

[6]  Thierry Gallouët,et al.  A finite volume scheme for anisotropic diffusion problems , 2004 .

[7]  Anton Arnold,et al.  Large-time behavior in non-symmetric Fokker-Planck equations , 2015, 1506.02470.

[8]  Claire Chainais-Hillairet,et al.  FINITE VOLUME APPROXIMATION FOR DEGENERATE DRIFT-DIFFUSION SYSTEM IN SEVERAL SPACE DIMENSIONS , 2004 .

[9]  M. Burger,et al.  A mixed finite element method for nonlinear diffusion equations , 2010 .

[10]  Francis Filbet,et al.  Asymptotic behaviour of a finite-volume scheme for the transient drift-diffusion model , 2007 .

[11]  Francis Filbet,et al.  Approximation of Hyperbolic Models for Chemosensitive Movement , 2005, SIAM J. Sci. Comput..

[12]  Claire Chainais-Hillairet,et al.  Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis , 2003 .

[13]  F. Filbet,et al.  On discrete functional inequalities for some finite volume schemes , 2012, 1202.4860.

[14]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[15]  Xiaoye S. Li,et al.  An overview of SuperLU: Algorithms, implementation, and user interface , 2003, TOMS.

[16]  Clément Cancès,et al.  Convergence of a nonlinear entropy diminishing Control Volume Finite Element scheme for solving anisotropic degenerate parabolic equations , 2015, Math. Comput..

[17]  Clément Cancès,et al.  Numerical Analysis of a Robust Free Energy Diminishing Finite Volume Scheme for Parabolic Equations with Gradient Structure , 2015, Found. Comput. Math..

[18]  C. Villani,et al.  Lyapunov functionals for boundary-driven nonlinear drift–diffusion equations , 2013, 1305.7405.

[19]  Evgueni A. Haroutunian,et al.  Information Theory and Statistics , 2011, International Encyclopedia of Statistical Science.

[20]  Benjamin Jourdain,et al.  Long-Time Asymptotics of a Multiscale Model for Polymeric Fluid Flows , 2006 .

[21]  François Bouchut,et al.  On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials , 1995, Differential and Integral Equations.

[22]  Giuseppe Toscani,et al.  ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS , 2001 .

[23]  Nader Masmoudi,et al.  Well‐posedness for the FENE dumbbell model of polymeric flows , 2008 .

[24]  Thierry Gallouët,et al.  A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension , 2006 .

[25]  Raphaèle Herbin,et al.  Small-stencil 3D schemes for diffusive flows in porous media , 2012 .

[26]  Maxime Herda,et al.  On massless electron limit for a multispecies kinetic system with external magnetic field , 2015, 1504.08139.

[27]  Clément Cancès,et al.  Numerical analysis of a robust entropy-diminishing Finite Volume scheme for parabolic equations with gradient structure , 2019 .

[28]  Francis Filbet,et al.  A Finite Volume Scheme for Nonlinear Degenerate Parabolic Equations , 2011, SIAM J. Sci. Comput..

[29]  Claire Chainais-Hillairet,et al.  ENTROPY-DISSIPATIVE DISCRETIZATION OF NONLINEAR DIFFUSION EQUATIONS AND DISCRETE BECKNER INEQUALITIES ∗ , 2013, 1303.3791.