On $τ$-preconditioners for a quasi-compact difference scheme to Riesz fractional diffusion equations with variable coefficients
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909171245383680 |
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| author | She, Zi-Hang Zhang, Xue Gu, Xian-Ming Serra-Capizzano, Stefano |
| author_facet | She, Zi-Hang Zhang, Xue Gu, Xian-Ming Serra-Capizzano, Stefano |
| contents | In the present study, we consider the numerical method for Toeplitz-like linear systems arising from the $d$-dimensional Riesz space fractional diffusion equations (RSFDEs). We apply the Crank-Nicolson (CN) technique to discretize the temporal derivative and apply a quasi-compact finite difference method to discretize the Riesz space fractional derivatives. For the $d$-dimensional problem, the corresponding coefficient matrix is the sum of a product of a (block) tridiagonal matrix multiplying a diagonal matrix and a $d$-level Toeplitz matrix. We develop a sine transform based preconditioner to accelerate the convergence of the GMRES method. Theoretical analyses show that the upper bound of relative residual norm of the preconditioned GMRES method with the proposed preconditioner is mesh-independent, which leads to a linear convergence rate. Numerical results are presented to confirm the theoretical results regarding the preconditioned matrix and to illustrate the efficiency of the proposed preconditioner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $τ$-preconditioners for a quasi-compact difference scheme to Riesz fractional diffusion equations with variable coefficients She, Zi-Hang Zhang, Xue Gu, Xian-Ming Serra-Capizzano, Stefano Numerical Analysis 34A08, 65F08, 65F10 In the present study, we consider the numerical method for Toeplitz-like linear systems arising from the $d$-dimensional Riesz space fractional diffusion equations (RSFDEs). We apply the Crank-Nicolson (CN) technique to discretize the temporal derivative and apply a quasi-compact finite difference method to discretize the Riesz space fractional derivatives. For the $d$-dimensional problem, the corresponding coefficient matrix is the sum of a product of a (block) tridiagonal matrix multiplying a diagonal matrix and a $d$-level Toeplitz matrix. We develop a sine transform based preconditioner to accelerate the convergence of the GMRES method. Theoretical analyses show that the upper bound of relative residual norm of the preconditioned GMRES method with the proposed preconditioner is mesh-independent, which leads to a linear convergence rate. Numerical results are presented to confirm the theoretical results regarding the preconditioned matrix and to illustrate the efficiency of the proposed preconditioner. |
| title | On $τ$-preconditioners for a quasi-compact difference scheme to Riesz fractional diffusion equations with variable coefficients |
| topic | Numerical Analysis 34A08, 65F08, 65F10 |
| url | https://arxiv.org/abs/2404.10221 |