High order asymptotic preserving scheme for linear kinetic equations with diffusive scaling

TL;DR

This paper develops high order asymptotic preserving numerical schemes for linear kinetic equations with diffusive scaling, capable of accurately capturing diffusive limits without stability constraints, and demonstrates their effectiveness through analysis and numerical tests.

Contribution

It introduces a novel high order time integrator framework for kinetic equations that preserves asymptotic limits and is free from diffusion stability restrictions.

Findings

Schemes are asymptotic preserving for various initial data.

Numerical results confirm high order accuracy in different regimes.

The methods effectively handle boundary conditions and degeneracies.

Abstract

In this work, high order asymptotic preserving schemes are constructed and analysed for kinetic equations under a diffusive scaling. The framework enables to consider different cases: the diffusion equation, the advection-diffusion equation and the presence of inflow boundary conditions. Starting from the micro-macro reformulation of the original kinetic equation, high order time integrators are introduced. This class of numerical schemes enjoys the Asymptotic Preserving (AP) property for arbitrary initial data and degenerates when $\epsilon$ goes to zero into a high order scheme which is implicit for the diffusion term, which makes it free from the usual diffusion stability condition. The space discretization is also discussed and high order methods are also proposed based on classical finite differences schemes. The Asymptotic Preserving property is analysed and numerical results are presented to illustrate the properties of the proposed schemes in different regimes.