Paired Explicit Relaxation Runge-Kutta Methods: Entropy-Conservative and Entropy-Stable High-Order Optimized Multirate Time Integration
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Doehring, Daniel Ranocha, Hendrik Torrilhon, Manuel |
| author_facet | Doehring, Daniel Ranocha, Hendrik Torrilhon, Manuel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_04991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Paired Explicit Relaxation Runge-Kutta Methods: Entropy-Conservative and Entropy-Stable High-Order Optimized Multirate Time Integration Doehring, Daniel Ranocha, Hendrik Torrilhon, Manuel Numerical Analysis Mathematical Physics 65L06, 65M20, 76-04, 70K20 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. |
| title | Paired Explicit Relaxation Runge-Kutta Methods: Entropy-Conservative and Entropy-Stable High-Order Optimized Multirate Time Integration |
| topic | Numerical Analysis Mathematical Physics 65L06, 65M20, 76-04, 70K20 |
| url | https://arxiv.org/abs/2507.04991 |