Multirate Time-Integration based on Dynamic ODE Partitioning through Adaptively Refined Meshes for Compressible Fluid Dynamics
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909243893874688 |
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| author | Doehring, Daniel Schlottke-Lakemper, Michael Gassner, Gregor J. Torrilhon, Manuel |
| author_facet | Doehring, Daniel Schlottke-Lakemper, Michael Gassner, Gregor J. Torrilhon, Manuel |
| contents | In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et. al. (2019, 2022) to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no changes in the spatial discretization methodology, making them readily implementable in existing codes that employ a method-of-lines approach.
We show that speedup compared to a range of state of the art Runge-Kutta methods can be realized, despite additional overhead due to the dynamic re-assignment of flagging variables and restricting nonlinear stability properties. The effectiveness of the approach is demonstrated for a range of simulation setups for viscous and inviscid convection-dominated compressible flows for which we provide a reproducibility repository.
In addition, we perform a thorough investigation of the nonlinear stability properties of the Paired-Explicit Runge-Kutta schemes regarding limitations due to the violation of monotonicity properties of the underlying spatial discretization. Furthermore, we present a novel approach for estimating the relevant eigenvalues of large Jacobians required for the optimization of stability polynomials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_05144 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multirate Time-Integration based on Dynamic ODE Partitioning through Adaptively Refined Meshes for Compressible Fluid Dynamics Doehring, Daniel Schlottke-Lakemper, Michael Gassner, Gregor J. Torrilhon, Manuel Numerical Analysis Mathematical Physics 65L06, 65M20, 76Mxx, 76Nxx In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et. al. (2019, 2022) to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no changes in the spatial discretization methodology, making them readily implementable in existing codes that employ a method-of-lines approach. We show that speedup compared to a range of state of the art Runge-Kutta methods can be realized, despite additional overhead due to the dynamic re-assignment of flagging variables and restricting nonlinear stability properties. The effectiveness of the approach is demonstrated for a range of simulation setups for viscous and inviscid convection-dominated compressible flows for which we provide a reproducibility repository. In addition, we perform a thorough investigation of the nonlinear stability properties of the Paired-Explicit Runge-Kutta schemes regarding limitations due to the violation of monotonicity properties of the underlying spatial discretization. Furthermore, we present a novel approach for estimating the relevant eigenvalues of large Jacobians required for the optimization of stability polynomials. |
| title | Multirate Time-Integration based on Dynamic ODE Partitioning through Adaptively Refined Meshes for Compressible Fluid Dynamics |
| topic | Numerical Analysis Mathematical Physics 65L06, 65M20, 76Mxx, 76Nxx |
| url | https://arxiv.org/abs/2403.05144 |