Convergence and error analysis of a semi-implicit finite volume scheme for the Gray--Scott system
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911121778147328 |
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| author | Randrianasolo, Tsiry Avisoa |
| author_facet | Randrianasolo, Tsiry Avisoa |
| contents | We analyze a semi-implicit finite volume scheme for the Gray--Scott system, a model for pattern formation in chemical and biological media. We prove unconditional well-posedness of the fully discrete problem and establish qualitative properties, including positivity and boundedness of the numerical solution. A convergence result is obtained by compactness arguments, showing that the discrete approximations converge strongly to a weak solution of the continuous system. Under additional regularity assumptions, we further derive a priori error estimates in the $L^2$ norm. Numerical experiments validate the theoretical analysis, confirm a rate of convergence of order 1, and illustrate the ability of the scheme to capture classical Gray--Scott patterns. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18910 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence and error analysis of a semi-implicit finite volume scheme for the Gray--Scott system Randrianasolo, Tsiry Avisoa Numerical Analysis 35Q92, 35K57, 65M08, 65M12, 65M15, 65M20 We analyze a semi-implicit finite volume scheme for the Gray--Scott system, a model for pattern formation in chemical and biological media. We prove unconditional well-posedness of the fully discrete problem and establish qualitative properties, including positivity and boundedness of the numerical solution. A convergence result is obtained by compactness arguments, showing that the discrete approximations converge strongly to a weak solution of the continuous system. Under additional regularity assumptions, we further derive a priori error estimates in the $L^2$ norm. Numerical experiments validate the theoretical analysis, confirm a rate of convergence of order 1, and illustrate the ability of the scheme to capture classical Gray--Scott patterns. |
| title | Convergence and error analysis of a semi-implicit finite volume scheme for the Gray--Scott system |
| topic | Numerical Analysis 35Q92, 35K57, 65M08, 65M12, 65M15, 65M20 |
| url | https://arxiv.org/abs/2508.18910 |