Improving Efficiency of Parallel Across the Method Spectral Deferred Corrections
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929708197740544 |
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| author | Čaklović, Gayatri Lunet, Thibaut Götschel, Sebastian Ruprecht, Daniel |
| author_facet | Čaklović, Gayatri Lunet, Thibaut Götschel, Sebastian Ruprecht, Daniel |
| contents | Parallel-across-the method time integration can provide small scale parallelism when solving initial value problems. Spectral deferred corrections (SDC) with a diagonal sweeper, which is closely related to iterated Runge-Kutta methods proposed by Van der Houwen and Sommeijer, can use a number of threads equal to the number of quadrature nodes in the underlying collocation method. However, convergence speed, efficiency and stability depends critically on the used coefficients. Previous approaches have used numerical optimization to find good parameters. Instead, we propose an ansatz that allows to find optimal parameters analytically. We show that the resulting parallel SDC methods provide stability domains and convergence order very similar to those of well established serial SDC variants. Using a model for computational cost that assumes 80% efficiency of an implementation of parallel SDC we show that our variants are competitive with serial SDC, previously published parallel SDC coefficients as well as Picard iteration, explicit RKM-4 and an implicit fourth-order diagonally implicit Runge-Kutta method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improving Efficiency of Parallel Across the Method Spectral Deferred Corrections Čaklović, Gayatri Lunet, Thibaut Götschel, Sebastian Ruprecht, Daniel Numerical Analysis Distributed, Parallel, and Cluster Computing 65R20, 65L04, 65L05, 65L20 G.1.7; G.1.8 Parallel-across-the method time integration can provide small scale parallelism when solving initial value problems. Spectral deferred corrections (SDC) with a diagonal sweeper, which is closely related to iterated Runge-Kutta methods proposed by Van der Houwen and Sommeijer, can use a number of threads equal to the number of quadrature nodes in the underlying collocation method. However, convergence speed, efficiency and stability depends critically on the used coefficients. Previous approaches have used numerical optimization to find good parameters. Instead, we propose an ansatz that allows to find optimal parameters analytically. We show that the resulting parallel SDC methods provide stability domains and convergence order very similar to those of well established serial SDC variants. Using a model for computational cost that assumes 80% efficiency of an implementation of parallel SDC we show that our variants are competitive with serial SDC, previously published parallel SDC coefficients as well as Picard iteration, explicit RKM-4 and an implicit fourth-order diagonally implicit Runge-Kutta method. |
| title | Improving Efficiency of Parallel Across the Method Spectral Deferred Corrections |
| topic | Numerical Analysis Distributed, Parallel, and Cluster Computing 65R20, 65L04, 65L05, 65L20 G.1.7; G.1.8 |
| url | https://arxiv.org/abs/2403.18641 |