Uniform accuracy of implicit-explicit Runge-Kutta (IMEX-RK) schemes for hyperbolic systems with relaxation

TL;DR

This paper proves that a class of IMEX-RK schemes maintain uniform stability and accuracy for hyperbolic systems with relaxation, regardless of the stiffness parameter, ensuring no order reduction occurs.

Contribution

The authors rigorously establish the uniform stability and accuracy of IMEX-RK schemes for hyperbolic systems with relaxation, demonstrating optimal performance independent of the stiffness parameter.

Findings

Uniform stability and accuracy proven for IMEX-RK schemes

No order reduction regardless of the stiffness parameter

Optimal order of accuracy maintained

Abstract

Implicit-explicit Runge-Kutta (IMEX-RK) schemes are popular methods to treat multiscale equations that contain a stiff part and a non-stiff part, where the stiff part is characterized by a small parameter $\varepsilon$. In this work, we prove rigorously the uniform stability and uniform accuracy of a class of IMEX-RK schemes for a linear hyperbolic system with stiff relaxation. The result we obtain is optimal in the sense that it holds regardless of the value of $\varepsilon$ and the order of accuracy is the same as the design order of the original scheme, i.e., there is no order reduction.