Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations

TL;DR

This paper analyzes semi-discrete Runge-Kutta schemes for nonlinear diffusion equations, establishing conditions for entropy dissipation linked to concavity properties, and verifies these conditions through theoretical analysis and numerical experiments.

Contribution

It introduces conditions ensuring entropy dissipation in semi-discrete Runge-Kutta schemes for nonlinear diffusion equations, connecting entropy concavity to geometric properties.

Findings

Conditions for local entropy dissipation are derived.

Entropy dissipation is shown to be global in numerical experiments.

Applicable to various equations including porous-medium and quantum diffusion.

Abstract

Semi-discrete Runge-Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps. The concavity property is shown to be related to the Bakry-Emery approach and the geodesic convexity of the entropy. The abstract conditions are verified for quasilinear parabolic equations (including the porous-medium equation), a linear diffusion system, and the fourth-order quantum diffusion equation. Numerical experiments for various Runge-Kutta finite-difference discretizations of the one-dimensional porous-medium equation show that the entropy-dissipation property is in fact global.