Exponential methods for solving hyperbolic problems with application to kinetic equations

TL;DR

This paper analyzes the stability issues of exponential integrators for kinetic equations and proposes Lawson methods as a more robust alternative, demonstrating their effectiveness through numerical simulations.

Contribution

It identifies stability deficiencies in classic exponential integrators and introduces Lawson methods as a more stable and efficient solution for kinetic equations.

Findings

Classic exponential integrators have stability issues.

Lawson methods do not suffer from these stability problems.

Numerical simulations confirm Lawson methods' efficiency.

Abstract

The efficient numerical solution of many kinetic models in plasma physics is impeded by the stiffness of these systems. Exponential integrators are attractive in this context as they remove the CFL condition induced by the linear part of the system, which in practice is often the most stringent stability constraint. In the literature, these schemes have been found to perform well, e.g., for drift-kinetic problems. Despite their overall efficiency and their many favorable properties, most of the commonly used exponential integrators behave rather erratically in terms of the allowed time step size in some situations. This severely limits their utility and robustness.Our goal in this paper is to explain the observed behavior and suggest exponential methods that do not suffer from the stated deficiencies. To accomplish this we study the stability of exponential integrators for a linearized problem. This analysis shows that classic exponential integrators exhibit severe deficiencies in that regard. Based on the analysis conducted we propose to use Lawson methods, which can be shown not to suffer from the same stability issues. We confirm these results and demonstrate the efficiency of Lawson methods by performing numerical simulations for both the Vlasov-Poisson system and a drift-kinetic model of a ion temperature gradient instability.