An optimal preconditioner for high-order scheme arising from multi-dimensional Riesz space fractional diffusion equations with variable coefficients
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908473767231488 |
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| author | Huang, Yuan-Yuan Qu, Wei Hon, Sean Y. Lei, Siu-Long |
| author_facet | Huang, Yuan-Yuan Qu, Wei Hon, Sean Y. Lei, Siu-Long |
| contents | In this paper, we propose an efficient method for solving multi-dimensional Riesz space fractional diffusion equations with variable coefficients. The Crank-Nicolson (CN) method is used for temporal discretization, while the fourth-order fractional centered difference (4FCD) method is employed for spatial discretization. Using a novel technique, we show that the CN-4FCD scheme for the multi-dimensional case is unconditionally stable and convergent, achieving second-order accuracy in time and fourth-order accuracy in space with respect to the discrete L2-norm. Moreover, leveraging the symmetric multi-level Toeplitz-like structure of the coefficient matrix in the discrete linear systems, we enhance the computational efficiency of the proposed scheme with a sine transform-based preconditioner, ensuring a mesh-size-independent convergence rate for the conjugate gradient method. Finally, two numerical examples validate the theoretical analysis and demonstrate the superior performance of the proposed preconditioner compared to existing methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23408 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An optimal preconditioner for high-order scheme arising from multi-dimensional Riesz space fractional diffusion equations with variable coefficients Huang, Yuan-Yuan Qu, Wei Hon, Sean Y. Lei, Siu-Long Numerical Analysis In this paper, we propose an efficient method for solving multi-dimensional Riesz space fractional diffusion equations with variable coefficients. The Crank-Nicolson (CN) method is used for temporal discretization, while the fourth-order fractional centered difference (4FCD) method is employed for spatial discretization. Using a novel technique, we show that the CN-4FCD scheme for the multi-dimensional case is unconditionally stable and convergent, achieving second-order accuracy in time and fourth-order accuracy in space with respect to the discrete L2-norm. Moreover, leveraging the symmetric multi-level Toeplitz-like structure of the coefficient matrix in the discrete linear systems, we enhance the computational efficiency of the proposed scheme with a sine transform-based preconditioner, ensuring a mesh-size-independent convergence rate for the conjugate gradient method. Finally, two numerical examples validate the theoretical analysis and demonstrate the superior performance of the proposed preconditioner compared to existing methods. |
| title | An optimal preconditioner for high-order scheme arising from multi-dimensional Riesz space fractional diffusion equations with variable coefficients |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2507.23408 |