On the Reynolds-number scaling of Poisson solver complexity
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909005976174592 |
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| author | Trias, F. Xavier Alsalti-Baldellou, Àdel Oliva, Assensi |
| author_facet | Trias, F. Xavier Alsalti-Baldellou, Àdel Oliva, Assensi |
| contents | We aim to answer the following question: is the complexity of numerically solving the Poisson equation increasing or decreasing for very large simulations of incompressible flows? Physical and numerical arguments are combined to derive power-law scalings at very high Reynolds numbers. A theoretical convergence analysis for both Jacobi and multigrid solvers defines a two-dimensional phase space divided into two regions depending on whether the number of solver iterations tends to decrease or increase with the Reynolds number. Numerical results indicate that, for Navier-Stokes turbulence, the complexity decreases with increasing Reynolds number, whereas for the one-dimensional Burgers equation it follows the opposite trend. The proposed theoretical framework thus provides a unified perspective on how solver convergence scales with the Reynolds number and offers valuable guidance for the development of next-generation preconditioning and multigrid strategies for extreme-scale simulations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22644 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Reynolds-number scaling of Poisson solver complexity Trias, F. Xavier Alsalti-Baldellou, Àdel Oliva, Assensi Fluid Dynamics Mathematical Physics Computational Physics We aim to answer the following question: is the complexity of numerically solving the Poisson equation increasing or decreasing for very large simulations of incompressible flows? Physical and numerical arguments are combined to derive power-law scalings at very high Reynolds numbers. A theoretical convergence analysis for both Jacobi and multigrid solvers defines a two-dimensional phase space divided into two regions depending on whether the number of solver iterations tends to decrease or increase with the Reynolds number. Numerical results indicate that, for Navier-Stokes turbulence, the complexity decreases with increasing Reynolds number, whereas for the one-dimensional Burgers equation it follows the opposite trend. The proposed theoretical framework thus provides a unified perspective on how solver convergence scales with the Reynolds number and offers valuable guidance for the development of next-generation preconditioning and multigrid strategies for extreme-scale simulations. |
| title | On the Reynolds-number scaling of Poisson solver complexity |
| topic | Fluid Dynamics Mathematical Physics Computational Physics |
| url | https://arxiv.org/abs/2512.22644 |