A high-order asymptotic-preserving scheme for kinetic equations using projective integration

TL;DR

This paper introduces a high-order explicit asymptotic-preserving scheme for kinetic equations that efficiently handles stiff components using projective integration, maintaining stability across different scalings.

Contribution

It develops a novel high-order scheme combining inner explicit steps with outer Runge-Kutta methods, ensuring stability and efficiency for kinetic equations in various regimes.

Findings

Inner step size can be chosen to match stability conditions of the limiting equation.

Number of inner steps is independent of the scaling parameter.

Numerical results confirm the scheme's stability and accuracy.

Abstract

We investigate a high-order, fully explicit, asymptotic-preserving scheme for a kinetic equation with linear relaxation, both in the hydrodynamic and diffusive scalings in which a hyperbolic, resp. parabolic, limiting equation exists. The scheme first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution and estimate the time derivative of the slow components. These estimated time derivatives are then used in an (outer) Runge-Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time-step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation. Moreover, the number of inner time steps is also independent of the scaling parameter. We analyse stability and consistency, and illustrate with numerical results.