Asymptotics and discretization of a weakly singular kernel: application to viscous flows in a network of thin tubes
Abstract
Kernels obtained from the heat equation arise in several modeling contexts, like some double porosity models, or viscous flows in networks of thin tubes. These kernels are weakly singular at initial time. An accurate approximation must therefore take this singularity into account. In this paper we obtain an asymptotic expansion for small times, which we use to build a numerical scheme for approximating the kernels. Convergence of the scheme and relevance of a correction through the asymptotics are proven both analytically and numerically. Finally, we show that this approximation applies to the model on the graph studied by the authors in Canon et al. doi:10.1016/j.jcp.2021.110262 (2021).
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