A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structure
This work provides a numerical method for a class of PDEs relevant to porous media, plasma physics, and polymer flows, ensuring correct long-time behavior.
The authors propose a finite volume scheme for nonlinear parabolic equations with non-homogeneous Dirichlet boundary conditions that preserve relative entropy structure. The scheme ensures exponential convergence to steady-state, with numerical results confirming accuracy and long-time asymptotic preservation.
We propose a finite volume scheme for a class of nonlinear parabolic equations endowed with non-homogeneous Dirichlet boundary conditions and which admit relative en-tropy 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.