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


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


  1. 1.
    Alpert, B.K.: Hybrid Gauss-trapezoidal quadrature rules. SIAM J. Sci. Comput. 20, 1551–1584 (1999). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Atkinson, K.E.: The numerical solution of integral equations of the second kind. Cambridge University Press, New York (1997)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bremer, J., Rokhlin, V., Sammis, I.: Universal quadratures for boundary integral equations on two-dimensional domains with corners. J. Comput. Phys. 229, 8259–8280 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Davis, P.J., Rabinowitz, P.: Methods of numerical integration. Academic Press, San Diego (1984)zbMATHGoogle Scholar
  5. 5.
    Greengard, L., Lee, J. -Y.: Stable and accurate integral equation methods for scattering problems with multiple interfaces in two dimensions. J. Comput. Phys. 231, 2389–2395 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Greengard, L., Lin, P.: Spectral approximation of the free-space heat kernel. Appl. Comput. Harmon. Anal. 9, 83–97 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Greengard, L., Strain, J.: A fast algorithm for the evaluation of heat potentials. Comm. Pure Appl. Math. 43, 949–963 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Greengard, L., Strain, J.: The fast Gauss transform. SIAM, J. Sci. Statist. Comput. 12, 79–94 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guenther, R.B., Lee, J.W.: Partial differential equations of mathematical physics and integral equations. Prentice Hall, Inglewood Cliffs (1988)Google Scholar
  10. 10.
    Helsing, J., Ojala, R.: Corner singularities for elliptic problems: integral equations, graded meshes, and compressed inverse preconditioning. J. Comput. Phys. 227, 8820–8840 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hille, E.: A class of reciprocal functions. Annals of Mathematics: Second Series 27, 427–464 (1926)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, J.-R., Greengard, L.: High order accurate methods for the evaluation of layer heat potentials. SIAM J. Sci. Comput. 31, 3847–3860 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lin, P.: On the numerical solution of the heat equation in unbounded domains. PhD thesis, New York University, New York (1993)Google Scholar
  14. 14.
    Pogorzelski, W.: Integral equations and their applications. Pergamon Press, Oxford (1966)zbMATHGoogle Scholar
  15. 15.
    Sampath, R.S., Sundar, H., Veerapaneni, S.: Parallel fast gauss transform. In: SC ’10: Proceedings of the ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis, New Orleans, LA, pp. 1–10 (2010)Google Scholar
  16. 16.
    Strain, J.: Fast adaptive methods for the free-space heat equation. SIAM J. Sci. Comput. 15, 185–206 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tausch, J.: A fast method for solving the heat equation by layer potentials, J. Comput. Phys. (2007)Google Scholar
  18. 18.
    Tausch, J., Weckiewicz, A.: Multidimensional fast Gauss transforms by Chebyshev expansions. SIAM J. Sci. Comput. 31, 3547–3565 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Trefethen, L.N.: Numerical computation of the Schwarz-Christoffel transformation. SIAM J. Sci. Stat. Comput. 1, 82–102 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, J., Greengard, L.: An adaptive fast Gauss transform in two dimensions. SIAM J. Sci. Comput. 40, A1274–A1300 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

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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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