Exponential Runge-Kutta methods for stiff kinetic equations

TL;DR

This paper presents exponential Runge-Kutta methods tailored for stiff kinetic equations, ensuring stability, positivity, and entropy preservation across a wide range of relaxation times, without solving nonlinear systems.

Contribution

Introduction of a new class of exponential Runge-Kutta methods that are unconditionally stable, asymptotic preserving, and applicable to both deterministic and probabilistic kinetic simulations.

Findings

Methods are exact for BGK relaxation operators.

They work uniformly for various relaxation times.

They preserve nonnegativity and entropy.

Abstract

We introduce a class of exponential Runge-Kutta integration methods for kinetic equations. The methods are based on a decomposition of the collision operator into an equilibrium and a non equilibrium part and are exact for relaxation operators of BGK type. For Boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes. We give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving. Such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times. The methods also offer favorable properties such as nonnegativity of the solution and entropy inequality. For this reason, as we will show, the methods are suitable both for deterministic as well as probabilistic numerical techniques.