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Journal of Scientific Computing

, Volume 79, Issue 2, pp 787–808 | Cite as

Fast High-Order Integral Equation Methods for Solving Boundary Value Problems of Two Dimensional Heat Equation in Complex Geometry

  • Shaobo Wang
  • Shidong Jiang
  • Jing WangEmail author
Article
  • 53 Downloads

Abstract

Efficient high-order integral equation methods have been developed for solving boundary value problems of the heat equation in complex geometry in two dimensions. First, the classical heat potential theory is applied to convert such problems to Volterra integral equations of the second kind via the heat layer potentials, where the unknowns are only on the space–time boundary. However, the heat layer potentials contain convolution integrals in both space and time whose direct evaluation requires \(O(N_S^2N_T^2)\) work and \(O(N_SN_T)\) storage, where \(N_S\) is the total number of discretization points on the spatial boundary and \(N_T\) is the total number of time steps. In order to evaluate the heat layer potentials accurately and efficiently, they are split into two parts—the local part containing the temporal integration from \(t-\delta \) to t and the history part containing the temporal integration from 0 to \(t-\delta \). The local part can be dealt with efficiently using conventional fast multipole type algorithms. For problems with complex stationary geometry, efficient separated sum-of-exponentials approximations are constructed for the heat kernel and used for the evaluation of the history part. Here all local and history kernels are compressed only once. The resulting algorithm is very efficient with quasilinear complexity in both space and time for both interior and exterior problems. For problems with complex moving geometry, the spectral Fourier approximation is applied for the heat kernel and nonuniform FFT is used to speed up the evaluation of the history part of heat layer potentials. The performance of both algorithms is demonstrated with several numerical examples.

Keywords

Heat equation Integral equation methods High-order methods Heat kernels Sum-of-exponentials approximation Nonuniform FFT 

Mathematics Subject Classification

30E15 35K05 35K08 45D05 65E05 65R10 80A20 

Notes

Acknowledgements

S. Jiang was supported by NSF under Grant DMS-1720405 and by the Flatiron Institute, a division of the Simons Foundation. Part of the work was done when J. Wang was visiting the Department of Mathematical Sciences at New Jersey Institute of Technology.

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Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNew Jersey Institute of TechnologyNewarkUSA
  2. 2.School of Information Science and EngineeringYunnan UniversityKunmingChina

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