An efficient preconditioner for evolutionary partial differential equations with $θ$-method in time discretization
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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_ | 1866917742881275904 |
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| author | Huang, Yuan-Yuan Fung, Po Yin Hon, Sean Y. Lin, Xue-Lei |
| author_facet | Huang, Yuan-Yuan Fung, Po Yin Hon, Sean Y. Lin, Xue-Lei |
| contents | In this study, the $θ$-method is used for discretizing a class of evolutionary partial differential equations. Then, we transform the resultant all-at-once linear system and introduce a novel one-sided preconditioner, which can be fast implemented in a parallel-in-time way. By introducing an auxiliary two-sided preconditioned system, we provide theoretical insights into the relationship between the residuals of the generalized minimal residual (GMRES) method when applied to both one-sided and two-sided preconditioned systems. Moreover, we show that the condition number of the two-sided preconditioned matrix is uniformly bounded by a constant that is independent of the matrix size, which in turn implies that the convergence behavior of the GMRES method for the one-sided preconditioned system is guaranteed. Numerical experiments confirm the efficiency and robustness of the proposed preconditioning approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03535 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An efficient preconditioner for evolutionary partial differential equations with $θ$-method in time discretization Huang, Yuan-Yuan Fung, Po Yin Hon, Sean Y. Lin, Xue-Lei Numerical Analysis 65F08, 65F10, 65M22, 15B05 In this study, the $θ$-method is used for discretizing a class of evolutionary partial differential equations. Then, we transform the resultant all-at-once linear system and introduce a novel one-sided preconditioner, which can be fast implemented in a parallel-in-time way. By introducing an auxiliary two-sided preconditioned system, we provide theoretical insights into the relationship between the residuals of the generalized minimal residual (GMRES) method when applied to both one-sided and two-sided preconditioned systems. Moreover, we show that the condition number of the two-sided preconditioned matrix is uniformly bounded by a constant that is independent of the matrix size, which in turn implies that the convergence behavior of the GMRES method for the one-sided preconditioned system is guaranteed. Numerical experiments confirm the efficiency and robustness of the proposed preconditioning approach. |
| title | An efficient preconditioner for evolutionary partial differential equations with $θ$-method in time discretization |
| topic | Numerical Analysis 65F08, 65F10, 65M22, 15B05 |
| url | https://arxiv.org/abs/2408.03535 |