Telescopic projective integration for kinetic equations with multiple relaxation times

TL;DR

This paper introduces a telescopic projective integration method tailored for efficiently solving stiff kinetic equations with multiple relaxation times, leveraging a hierarchy of projective levels to handle multiple eigenvalue clusters.

Contribution

The paper extends projective integration to telescopic methods for kinetic equations with multiple relaxation times, improving efficiency and stability in stiff regimes.

Findings

Number of projective levels depends on eigenvalue clusters.

Outer time step size depends only on the slowest time scale.

Method successfully applied to 1D and 2D simulations.

Abstract

We study a general, high-order, fully explicit numerical method for simulating kinetic equations with a BGK-type collision model with multiple relaxation times. In that case, the problem is stiff and its spectrum consists of multiple separated eigenvalue clusters. Projective integration methods are explicit integration schemes that first take a few small (inner) steps with a simple, explicit method, after which the solution is extrapolated forward in time over a large (outer) time step. These are very efficient schemes, provided there are only two clusters of eigenvalues. Telescopic projective integration methods generalize the idea of projective integration methods by constructing a hierarchy of projective levels. Here, we show how telescopic projective integration methods can be used to efficiently integrate kinetic equations with multiple relaxation times. We show that the required number of projective levels depends on the number of clusters, which in turn depends on the stiffness of the BGK source term. The size of the outer level time step only depends on the slowest time scale present in the model and is independent of the stiffness of the problem. We discuss stability and illustrate the approach with simulations in one and two spatial dimensions.