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arXiv · 2506.18649

Semi-discrete heat equations with variable coefficients and the parametrix method

Ulrik S. Fjordholm·Kenneth H. Karlsen·Peter H. C. Pang
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Abstract

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.

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Ulrik S. Fjordholm, Kenneth H. Karlsen, Peter H. C. Pang. 2025-06-23. Semi-discrete heat equations with variable coefficients and the parametrix method. https://doi.org/10.1093/imanum%2Fdrag004

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