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Advances in Computational Mathematics

, Volume 45, Issue 2, pp 847–867 | Cite as

Hybrid asymptotic/numerical methods for the evaluation of layer heat potentials in two dimensions

  • Jun WangEmail author
  • Leslie Greengard
Article
  • 40 Downloads

Abstract

We present a hybrid asymptotic/numerical method for the accurate computation of single- and double-layer heat potentials in two dimensions. It has been shown in previous work that simple quadrature schemes suffer from a phenomenon called “geometrically induced stiffness,” meaning that formally high-order accurate methods require excessively small time steps before the rapid convergence rate is observed. This can be overcome by analytic integration in time, requiring the evaluation of a collection of spatial boundary integral operators with non-physical, weakly singular kernels. In our hybrid scheme, we combine a local asymptotic approximation with the evaluation of a few boundary integral operators involving only Gaussian kernels, which are easily accelerated by a new version of the fast Gauss transform. This new scheme is robust, avoids geometrically induced stiffness, and is easy to use in the presence of moving geometries. Its extension to three dimensions is natural and straightforward, and should permit layer heat potentials to become flexible and powerful tools for modeling diffusion processes.

Keywords

Hybrid asymptotic/numerical method Geometrically induced stiffness Gauss transform 

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Notes

Acknowledgments

We would like to thank Alex Barnett and Shidong Jiang for several useful conversations.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Flatiron Institute, Simons FoundationNew YorkUSA

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