Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit

TL;DR

This paper introduces an asymptotic-preserving projective integration scheme for kinetic equations approaching diffusion limits, enabling stable large time steps independent of mean free path, with proven consistency and demonstrated effectiveness.

Contribution

It develops a novel projective integration method that remains stable and consistent in the diffusion limit, independent of the mean free path, and provides theoretical and numerical validation.

Findings

Time-step restriction matches diffusion stability conditions.

Number of inner steps is independent of mean free path.

Method converges to standard diffusion scheme in the limit.

Abstract

We investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path, in which the kinetic description approaches a diffusion phenomenon. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large time step on the diffusion time scale. We show that, with an appropriate choice of the inner step size, the time-step restriction on the outer time step is similar to the stability condition for the diffusion equation, whereas the required number of inner steps does not depend on the mean free path. We also provide a consistency result. The presented method is asymptotic-preserving, in the sense that the method converges to a standard finite volume scheme for the diffusion equation in the limit of vanishing mean free path. The analysis is illustrated with numerical results, and we present an application to the Su-Olson test.