Structure-preserving preconditioning of discrete space-fractional diffusion equations with variable coefficient and θ-Method
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911519405506560 |
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| author | Khan, Muhammad Faisal Ilyas, Asim Krause, Rolf Serra-Capizzano, Stefano Tablino-Possio, Cristina |
| author_facet | Khan, Muhammad Faisal Ilyas, Asim Krause, Rolf Serra-Capizzano, Stefano Tablino-Possio, Cristina |
| contents | This paper studies the spectral properties of large matrices and the preconditioning of linear systems, arising from the finite difference discretization of a time-dependent space-fractional diffusion equation with a variable coefficient $a(x)$ defined on $Ω\subset \mathbb{R}^d$, $d=1,2$. The model involves a one-sided Riemann-Liouville fractional derivative multiplied by the function $a(x)$, discretized by the shifted Gr"unwald formula in space and the $θ$-method in time. The resulting all-at-once linear systems exhibit a $(d+1)$-level Toeplitz-like matrix structure, with $d=1,2$ denoting the space dimension, while the additional level is due to the time variable.
A preconditioning strategy is developed based on the structural properties of the discretized operator. Using the generalized locally Toeplitz (GLT) theory, we analyze the spectral distribution of the unpreconditioned and preconditioned matrix sequences. The main novelty is that the analysis fully covers the case where the variable coefficient $a$ is nonconstant. Numerical results are provided to support the GLT based theoretical findings, and some possible extensions are briefly discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15122 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structure-preserving preconditioning of discrete space-fractional diffusion equations with variable coefficient and θ-Method Khan, Muhammad Faisal Ilyas, Asim Krause, Rolf Serra-Capizzano, Stefano Tablino-Possio, Cristina Numerical Analysis This paper studies the spectral properties of large matrices and the preconditioning of linear systems, arising from the finite difference discretization of a time-dependent space-fractional diffusion equation with a variable coefficient $a(x)$ defined on $Ω\subset \mathbb{R}^d$, $d=1,2$. The model involves a one-sided Riemann-Liouville fractional derivative multiplied by the function $a(x)$, discretized by the shifted Gr"unwald formula in space and the $θ$-method in time. The resulting all-at-once linear systems exhibit a $(d+1)$-level Toeplitz-like matrix structure, with $d=1,2$ denoting the space dimension, while the additional level is due to the time variable. A preconditioning strategy is developed based on the structural properties of the discretized operator. Using the generalized locally Toeplitz (GLT) theory, we analyze the spectral distribution of the unpreconditioned and preconditioned matrix sequences. The main novelty is that the analysis fully covers the case where the variable coefficient $a$ is nonconstant. Numerical results are provided to support the GLT based theoretical findings, and some possible extensions are briefly discussed. |
| title | Structure-preserving preconditioning of discrete space-fractional diffusion equations with variable coefficient and θ-Method |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2603.15122 |