Paired Explicit Relaxation Runge-Kutta Methods: Entropy-Conservative and Entropy-Stable High-Order Optimized Multirate Time Integration

TL;DR

This paper introduces new entropy-conservative and entropy-stable multirate Runge-Kutta methods with relaxation, optimized up to fourth-order, demonstrating improved stability and efficiency for complex PDE systems on unstructured meshes.

Contribution

It develops novel Paired Explicit Relaxation Runge-Kutta methods that enhance stability and efficiency for conservation laws and PDEs, with practical implementation and validation.

Findings

Methods achieve up to fourth-order accuracy.

Enhanced nonlinear stability and robustness.

Outperform standalone methods by up to a factor of four.

Abstract

We present novel entropy-conservative and entropy-stable multirate Runge-Kutta methods based on Paired Explicit Runge-Kutta (P-ERK) schemes with relaxation for conservation laws and related systems of partial differential equations. Optimized schemes up to fourth-order of accuracy are derived and validated in terms of order of consistency, conservation of linear invariants, and entropy conservation/stability. We demonstrate the effectiveness of these P-ERRK methods when combined with a high-order, entropy-conservative/stable discontinuous Galerkin spectral element method on unstructured meshes. The Paired Explicit Relaxation Runge-Kutta methods(P-ERRK) are readily implemented for partitioned semidiscretizations arising from problems with equation-based scale separation such as non-uniform meshes. We highlight that the relaxation approach acts as a time-limiting technique which improves the nonlinear stability and thus robustness of the multirate schemes. The P-ERRK methods are applied to a range of problems, ranging from compressible Euler over compressible Navier-Stokes to the visco-resistive magnetohydrodynamics equations in two and three spatial dimensions. For each test case, we compare computational load and runtime to standalone relaxed Runge-Kutta methods which are outperformed by factors up to four. All results can be reproduced using a publicly available repository.