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

Abstract

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.

Large-time behaviour of a family of finite volume schemes for boundary-driven convection–diffusion equations

“…In this paper, we show that the numerical analysis performed in [24] and the adaptation of the entropy method of [7] also works for some usual and well-known schemes. These schemes, which include the upwind, centered and Scharfetter-Gummel scheme, do not need any precalculation of the steady-state, contrary to the schemes of [24]. In the case of Fokker-Plank or porous media equations, we prove exponential decay towards the steady-state defined by each scheme.…”

“…We refer to [7,Theorem 1.4] or [24,Proposition 1.3] for a proof. Typical examples of relative φ-entropies are the physical relative entropy and p-entropies (or Tsallis relative entropies) respectively generated, by…”

“…The question of the large time behavior of numerical schemes has been investigated for instance for coagulationfragmentation models [23], for nonlinear diffusion equations [15], for reaction-diffusion systems [29,30], for drift-diffusion systems [14,5]. In [24], Filbet and Herda proposed a finite volume scheme preserving entropy principles for nonlinear boundary-driven Fokker-Planck equations. They adapted the entropy method from [7] to the discrete level by defining a finite volume approximation of the steady-state and using it to design a scheme for the evolution equation.…”

Energy stable numerical methods for porous media flow type problems

“…This question is indeed of broad interest for porous media flows, in particular in the context where the time scales are long like for instance in basin modeling or for nuclear waste repository management. Therefore, the study of the long-time behavior of Finite Volume schemes for convection-diffusion problems has been the purpose of several contributions (see, e.g., [13,62,[95][96][97]).…”

Analytical solution of stochastic differential equation by multilayer perceptron neural network approximation of Fokker–Planck equation

“…They can be used to solve a wide variety of phenomena such as sound, heat, fluid dynamics [5][6][7]. Much research has been devoted to find the numerical solution of FPE, different standard methods for solving PDEs such as finite difference, finite volume, finite element, and spectral methods have been employed to solve FPE [8][9][10][11][12][13][14][15][16][17]. In a different approach, the solution of the FPE was approximated by RBF networks [18].…”