Hybrid Asymptotic/Numerical Methods for the Evaluation of Layer Heat Potentials in Two Dimensions

Jun Wang Yau Mathematical Sciences Center, Tsinghua University Leslie Greengard Courant Institute, New York University

Numerical Analysis and Scientific Computing mathscidoc:2205.25002

Advances in Computational Mathematics, 2018.10
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.
Hybrid asymptotic/numerical method , Geometrically induced stiffness, Gauss transform
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@inproceedings{jun2018hybrid,
  title={Hybrid Asymptotic/Numerical Methods for the Evaluation of Layer Heat Potentials in Two Dimensions },
  author={Jun Wang, and Leslie Greengard},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517121728831770186},
  booktitle={Advances in Computational Mathematics},
  year={2018},
}
Jun Wang, and Leslie Greengard. Hybrid Asymptotic/Numerical Methods for the Evaluation of Layer Heat Potentials in Two Dimensions . 2018. In Advances in Computational Mathematics. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517121728831770186.
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