Implicit-explicit linear multistep methods for stiff kinetic equations

TL;DR

This paper introduces high-order asymptotic-preserving linear multistep methods for stiff kinetic equations, demonstrating advantages over IMEX Runge-Kutta schemes in efficiency and simplicity, especially for multidimensional problems.

Contribution

The paper develops and extends high-order IMEX multistep methods for kinetic equations, including the full Boltzmann equation, with analysis of their behavior in the Navier-Stokes regime.

Findings

IMEX multistep schemes are more computationally efficient than IMEX Runge-Kutta methods.

The methods are asymptotic-preserving and suitable for multidimensional kinetic equations.

Compatibility conditions in the Navier-Stokes regime are derived.

Abstract

We consider the development of high order asymptotic-preserving linear multistep methods for kinetic equations and related problems. The methods are first developed for BGK-like kinetic models and then extended to the case of the full Boltzmann equation. The behavior of the schemes in the Navier-Stokes regime is also studied and compatibility conditions derived. We show that, compared to IMEX Runge-Kutta methods, the IMEX multistep schemes have several advantages due to the absence of coupling conditions and to the greater computational efficiency. The latter is of paramount importance when dealing with the time discretization of multidimensional kinetic equations.