A parallel preconditioner for the all-at-once linear system from evolutionary PDEs with Crank-Nicolson discretization
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916120959647744 |
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| author | Zhao, Yong-Liang Gu, Xian-Ming Oosterlee, Cornelis W. |
| author_facet | Zhao, Yong-Liang Gu, Xian-Ming Oosterlee, Cornelis W. |
| contents | The Crank-Nicolson (CN) method is a well-known time integrator for evolutionary partial differential equations (PDEs) arising in many real-world applications. Since the solution at any time depends on the solution at previous time steps, the CN method is inherently difficult to parallelize. In this paper, we consider a parallel method for the solution of evolutionary PDEs with the CN scheme. Using an all-at-once approach, we can solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method. Due to the diagonalization of the proposed preconditioner, we can prove that most eigenvalues of preconditioned matrices are equal to 1 and the others lie in the set: $\left\{z\in\mathbb{C}: 1/(1 + α) < |z| < 1/(1 - α)~{\rm and}~\Re{\rm e}(z) > 0\right\}$, where $0 < α< 1$ is a free parameter. Besides, the efficient implementation of the proposed preconditioner is described. Given certain conditions, we prove that the preconditioned GMRES method exhibits a mesh-independent convergence rate. Finally, we will verify both theoretical findings and the efficacy of the proposed preconditioner via numerical experiments on financial option pricing PDEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16113 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A parallel preconditioner for the all-at-once linear system from evolutionary PDEs with Crank-Nicolson discretization Zhao, Yong-Liang Gu, Xian-Ming Oosterlee, Cornelis W. Numerical Analysis 65L05, 65N22, 65F10 The Crank-Nicolson (CN) method is a well-known time integrator for evolutionary partial differential equations (PDEs) arising in many real-world applications. Since the solution at any time depends on the solution at previous time steps, the CN method is inherently difficult to parallelize. In this paper, we consider a parallel method for the solution of evolutionary PDEs with the CN scheme. Using an all-at-once approach, we can solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method. Due to the diagonalization of the proposed preconditioner, we can prove that most eigenvalues of preconditioned matrices are equal to 1 and the others lie in the set: $\left\{z\in\mathbb{C}: 1/(1 + α) < |z| < 1/(1 - α)~{\rm and}~\Re{\rm e}(z) > 0\right\}$, where $0 < α< 1$ is a free parameter. Besides, the efficient implementation of the proposed preconditioner is described. Given certain conditions, we prove that the preconditioned GMRES method exhibits a mesh-independent convergence rate. Finally, we will verify both theoretical findings and the efficacy of the proposed preconditioner via numerical experiments on financial option pricing PDEs. |
| title | A parallel preconditioner for the all-at-once linear system from evolutionary PDEs with Crank-Nicolson discretization |
| topic | Numerical Analysis 65L05, 65N22, 65F10 |
| url | https://arxiv.org/abs/2401.16113 |