A finite volume scheme for nonlinear degenerate parabolic equations

TL;DR

This paper introduces a second-order finite volume scheme for nonlinear degenerate parabolic equations that accurately preserves steady-states and demonstrates high-order accuracy and efficiency in long-time simulations across degenerate and non-degenerate regimes.

Contribution

The paper presents a novel second-order finite volume scheme that maintains steady-states and achieves high accuracy for nonlinear degenerate parabolic equations, including in degenerate cases.

Findings

Scheme preserves steady-states effectively.

High-order accuracy confirmed in numerical tests.

Efficient long-time behavior simulation demonstrated.

Abstract

We propose a second order finite volume scheme for nonlinear degenerate parabolic equations. For some of these models (porous media equation, drift-diffusion system for semiconductors, ...) it has been proved that the transient solution converges to a steady-state when time goes to infinity. The present scheme preserves steady-states and provides a satisfying long-time behavior. Moreover, it remains valid and second-order accurate in space even in the degenerate case. After describing the numerical scheme, we present several numerical results which confirm the high-order accuracy in various regime degenerate and non degenerate cases and underline the efficiency to preserve the large-time asymptotic.