Long-time behavior of a finite volume discretization for a fourth order diffusion equation

TL;DR

This paper introduces a finite-volume scheme for a nonlinear fourth order diffusion equation that preserves key structural properties, enabling accurate long-term behavior analysis and capturing complex dynamics effectively.

Contribution

The paper presents a novel finite-volume discretization that maintains the gradient flow structure and related properties of the continuous equation.

Findings

The scheme preserves entropy and Fisher information as Lyapunov functionals.

Dissipation rates converge to optimal continuous limits.

Numerical experiments show effective capture of complex dynamics.

Abstract

We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the $d$-dimensional cube, for arbitrary $d \geq 1$. The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker-Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.