All-stages-implicit and strong-stability-preserving implicit-explicit Runge-Kutta time discretization schemes for hyperbolic systems with stiff relaxation terms

TL;DR

This paper introduces eight all-stages-implicit IMEX Runge-Kutta schemes that are strongly stability-preserving and suitable for hyperbolic systems with stiff relaxation, offering improved stability and accuracy in the zero relaxation limit.

Contribution

The paper develops new all-stages-implicit IMEX Runge-Kutta schemes with SSP properties, capable of handling stiff relaxation terms uniformly and accurately.

Findings

Eight new IMEX RK schemes constructed up to third order.

Some schemes are SSP for both explicit and implicit parts.

Schemes recover accuracy and converge uniformly under certain conditions.

Abstract

We construct eight implicit-explicit (IMEX) Runge-Kutta (RK) schemes up to third order of the type in which all stages are implicit so that they can be used in the zero relaxation limit in a unified and convenient manner. These all-stages-implicit (ASI) schemes attain the strong-stability-preserving (SSP) property in the limiting case, and two are SSP for not only the explicit part but also the implicit part and the entire IMEX scheme. Three schemes can completely recover to the designed accuracy order in two sides of the relaxation parameter for both equilibrium and non-equilibrium initial conditions. Two schemes converge nearly uniformly for equilibrium cases. These ASI schemes can be used for hyperbolic systems with stiff relaxation terms or differential equations with some type constraints.