Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit

TL;DR

This paper develops and analyzes Implicit-Explicit Runge-Kutta schemes for hyperbolic systems with stiff relaxation, ensuring accurate diffusion limit behavior and improved stability for convection-diffusion equations.

Contribution

It introduces a reformulation of IMEX R-K schemes that naturally transition to schemes suitable for the diffusion limit, maintaining stability and accuracy.

Findings

Schemes reduce to explicit schemes in the diffusion limit without modification.

The proposed approach handles stiff relaxation effectively in hyperbolic systems.

Numerical examples confirm theoretical stability and accuracy improvements.

Abstract

We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such regime the system relaxes towards a convection-diffusion equation. The first objective of the paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of the paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by an original reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis on such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, under the same conditions derived for hyperbolic relaxation. Several numerical examples including neutron transport equations confirm the theoretical analysis.