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

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 (PDEs). 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 Paired Explicit Relaxation Runge-Kutta (P-ERRK) methods when combined with a high-order, entropy-conservative/stable discontinuous Galerkin spectral element method on unstructured meshes. The P-ERRK methods 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 (VRMHD) 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.