Semi-discrete heat equations with variable coefficients and the parametrix method
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909985227669504 |
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| author | Fjordholm, Ulrik S. Karlsen, Kenneth H. Pang, Peter H. C. |
| author_facet | Fjordholm, Ulrik S. Karlsen, Kenneth H. Pang, Peter H. C. |
| contents | We develop a parametrix approach for constructing solutions and establishing grid-size independent estimates for semi-discrete heat equations with variable coefficients. While the classical continuous setting benefits from Gaussian estimates of the constant coefficient heat kernel, such estimates are not available in the semi-discrete context. To address this complication, we derive estimates involving products of heavy-tailed Lorentz (also known as Cauchy) probability densities. These Lorentzian estimates provide a sufficient handle on certain iterated convolutions involving Bessel functions, enabling us to achieve convergence of the parametrix approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18649 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semi-discrete heat equations with variable coefficients and the parametrix method Fjordholm, Ulrik S. Karlsen, Kenneth H. Pang, Peter H. C. Numerical Analysis 35K15, 35K08 (Primary) 65M06, 33C10 (Secondary) We develop a parametrix approach for constructing solutions and establishing grid-size independent estimates for semi-discrete heat equations with variable coefficients. While the classical continuous setting benefits from Gaussian estimates of the constant coefficient heat kernel, such estimates are not available in the semi-discrete context. To address this complication, we derive estimates involving products of heavy-tailed Lorentz (also known as Cauchy) probability densities. These Lorentzian estimates provide a sufficient handle on certain iterated convolutions involving Bessel functions, enabling us to achieve convergence of the parametrix approach. |
| title | Semi-discrete heat equations with variable coefficients and the parametrix method |
| topic | Numerical Analysis 35K15, 35K08 (Primary) 65M06, 33C10 (Secondary) |
| url | https://arxiv.org/abs/2506.18649 |