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

2025/06/23 by Fjordholm, Ulrik S., Karlsen, Kenneth H., Pang, Peter H. C.
#33C10 (Secondary) #35K08 (Primary) 65M06 #35K15 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2506.18649

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.

Citations

• Constructing heat kernels on infinite graphs
• The parametrix construction of the heat kernel on a graph
• Gaussian estimates for symmetric simple exclusion processes

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