An entropic method for discrete systems with Gibbs entropy

TL;DR

This paper introduces an entropic correction scheme for numerical integration of systems with Gibbs entropy, ensuring entropy monotonicity without significantly affecting the original scheme's accuracy, validated through experiments on Fokker-Planck and Boltzmann equations.

Contribution

It proposes a simple entropy fix method applicable after each time step that preserves entropy properties with minimal impact on numerical order.

Findings

The entropy fix maintains the order of the original scheme.

Numerical experiments confirm the scheme's effectiveness on Fokker-Planck and Boltzmann equations.

The method is broadly applicable to systems with Gibbs entropy.

Abstract

We consider general systems of ordinary differential equations with monotonic Gibbs entropy, and introduce an entropic scheme that simply imposes an entropy fix after every time step of any existing time integrator. It is proved that in the general case, our entropy fix has only infinitesimal influence on the numerical order of the original scheme, and in many circumstances, it can be shown that the scheme does not affect the numerical order. Numerical experiments on the linear Fokker-Planck equation and nonlinear Boltzmann equation are carried out to support our numerical analysis.