Fourth-Order Paired-Explicit Runge-Kutta Methods
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911204660740096 |
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| author | Doehring, Daniel Christmann, Lars Schlottke-Lakemper, Michael Gassner, Gregor J. Torrilhon, Manuel |
| author_facet | Doehring, Daniel Christmann, Lars Schlottke-Lakemper, Michael Gassner, Gregor J. Torrilhon, Manuel |
| contents | In this paper, we extend the Paired-Explicit Runge-Kutta schemes by Vermeire et. al. to fourth-order of consistency. Based on the order conditions for partitioned Runge-Kutta methods we motivate a specific form of the Butcher arrays which leads to a family of fourth-order accurate methods. The employed form of the Butcher arrays results in a special structure of the stability polynomials, which needs to be adhered to for an efficient optimization of the domain of absolute stability. We demonstrate that the constructed fourth-order Paired-Explicit Runge-Kutta methods satisfy linear stability, internal consistency, designed order of convergence, and conservation of linear invariants. At the same time, these schemes are seamlessly coupled for codes employing a method-of-lines approach, in particular without any modifications of the spatial discretization. We apply the multirate Paired-Explicit Runge-Kutta (P-ERK) schemes to inviscid and viscous problems with locally varying wave speeds, which may be induced by non-uniform grids or multiscale properties of the governing partial differential equation. Compared to state-of-the-art optimized standalone methods, the multirate P-ERK schemes allow significant reductions in right-hand-side evaluations and wall-clock time, ranging from 40% up to factors greater than three. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_05470 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fourth-Order Paired-Explicit Runge-Kutta Methods Doehring, Daniel Christmann, Lars Schlottke-Lakemper, Michael Gassner, Gregor J. Torrilhon, Manuel Numerical Analysis Mathematical Physics 65L06, 65M20, 7604 In this paper, we extend the Paired-Explicit Runge-Kutta schemes by Vermeire et. al. to fourth-order of consistency. Based on the order conditions for partitioned Runge-Kutta methods we motivate a specific form of the Butcher arrays which leads to a family of fourth-order accurate methods. The employed form of the Butcher arrays results in a special structure of the stability polynomials, which needs to be adhered to for an efficient optimization of the domain of absolute stability. We demonstrate that the constructed fourth-order Paired-Explicit Runge-Kutta methods satisfy linear stability, internal consistency, designed order of convergence, and conservation of linear invariants. At the same time, these schemes are seamlessly coupled for codes employing a method-of-lines approach, in particular without any modifications of the spatial discretization. We apply the multirate Paired-Explicit Runge-Kutta (P-ERK) schemes to inviscid and viscous problems with locally varying wave speeds, which may be induced by non-uniform grids or multiscale properties of the governing partial differential equation. Compared to state-of-the-art optimized standalone methods, the multirate P-ERK schemes allow significant reductions in right-hand-side evaluations and wall-clock time, ranging from 40% up to factors greater than three. |
| title | Fourth-Order Paired-Explicit Runge-Kutta Methods |
| topic | Numerical Analysis Mathematical Physics 65L06, 65M20, 7604 |
| url | https://arxiv.org/abs/2408.05470 |