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


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.


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 



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.


  1. 1.
    Alpert, B.K.: Hybrid Gauss-trapezoidal quadrature rules. SIAM J. Sci. Comput. 20(5), 1551–1584 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barnett, A., Magland, J.: Non-uniform fast Fourier transform library of types 1, 2, 3 in dimensions 1, 2, 3. (2018)
  3. 3.
    Brattkus, K., Meiron, D.I.: Numerical simulations of unsteady crystal growth. SIAM J. Appl. Math. 52, 1303–1320 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bremer, J.: A fast direct solver for the integral equations of scattering theory on planar curves with corners. J. Comput. Phys. 231(4), 1879–1899 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brown, M.: The method of layer potentials for the heat equation in Lipschitz cylinders. Am. J. Math. 111, 339–379 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brown, M.: The initial-Neumann problem for the heat equation in Lipschitz cylinders. Trans. Am. Math. Soc. 320, 1–52 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chandrasekaran, S., Dewilde, P., Gu, M., Lyons, W., Pals, T.: A fast solver for HSS representations via sparse matrices. SIAM J. Matrix Anal. Appl. 29, 67–81 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cheng, H., Greengard, L., Rokhlin, V.: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155(2), 468–498 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fabes, E.B., Riviere, N.M.: Dirichlet and Neumann problems for the heat equation in \(c1\) cylinders. Proc. Sympos. Pure Math. 35, 179–196 (1979)CrossRefGoogle Scholar
  10. 10.
    Fong, W., Darve, E.: The black-box fast multipole method. J. Comput. Phys. 228(23), 8712–8725 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gimbutas, Z., Rokhlin, V.: A generalized fast multipole method for nonoscillatory kernels. SIAM J. Sci. Comput. 24, 796–817 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Greengard, L., Lee, J.: Accelerating the nonuniform fast Fourier transform. SIAM Rev. 46, 443–454 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Greengard, L., Lin, P.: Spectral approximation of the free-space heat kernel. Appl. Comput. Harmon. Anal. 9, 83–97 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. Acta. Numer. 6, 229–270 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Greengard, L., Strain, J.: A fast algorithm for the evaluation of heat potentials. Commun. Pure Appl. Math. 43, 949–963 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Greengard, L., Strain, J.: The fast Gauss transform. SIAM J. Sci. Statist. Comput. 12, 79–94 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Greengard, L., Sun, X.: A new version of the fast Gauss transform. Doc. Math. III, 575–584 (1990)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Guenther, R.B., Lee, J.W.: Partial Differential Equations of Mathematical Physics and Integral Equations. Prentice-Hall, Englewood Cliffs (1988)Google Scholar
  20. 20.
    Hackbusch, W., Börm, S.: Data-sparse approximation by adaptive H2-matrices. Computing 69(1), 1–35 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ho, K.L., Greengard, L.: A fast direct solver for structured linear systems by recursive skeletonization. SIAM J. Sci. Comput. 34(5), A2507–A2532 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ho, K.L., Ying, L.: Hierarchical interpolative factorization for elliptic operators: integral equations. Commun. Pure Appl. Math. 69(7), 1314–1353 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ibanez, M.T., Power, H.: An efficient direct BEM numerical scheme for phase change problems using Fourier series. Comput. Methods Appl. Mech. Eng. 191, 2371–2402 (2002)CrossRefzbMATHGoogle Scholar
  24. 24.
    Jiang, S., Greengard, L., Wang, S.: Efficient sum-of-exponentials approximations for the heat kernel and their applications. Adv. Comput. Math. 41(3), 529–551 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Jiang, S., Rachh, M., Xiang, Y.: An efficient high order method for dislocation climb in two dimensions. SIAM J. Multi. Model. Simul. 15(1), 235–253 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jiang, S., Veerapaneni, S., Greengard, L.: Integral equation methods for unsteady Stokes flow in two dimensions. SIAM J. Sci. Comput. 34(4), A2197–A2219 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Jiang, S., Wang, S.: An efficient high-order integral equation method for solving the heat equation with complex geometries in three dimensions. In: Proceedings of the 7th ICCM (2019) (in press)Google Scholar
  28. 28.
    Kong, W.Y., Bremer, J., Rokhlin, V.: An adaptive fast direct solver for boundary integral equations in two dimensions. Appl. Comput. Harmon. Anal. 31(3), 346–369 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kress, R.: Linear integral equations. In: Applied Mathematical Sciences, vol. 82, 3rd edn. Springer, Berlin (2014)Google Scholar
  30. 30.
    Lee, J.Y., Greengard, L., Gimbutas, Z.: NUFFT Version 1.3.2 Software Release. (2009)
  31. 31.
    Li, J., Greengard, L.: On the numerical solution of the heat equation. I. Fast solvers in free space. J. Comput. Phys. 226(2), 1891–1901 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Li, J., Greengard, L.: High order accurate methods for the evaluation of layer heat potentials. SIAM J. Sci. Comput. 31, 3847–3860 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Martinsson, P.G.: A fast direct solver for a class of elliptic partial differential equations. J. Sci. Comput. 38(3), 316–330 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Martinsson, P.G., Rokhlin, V.: A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys. 205(1), 1–23 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Martinsson, P.G., Rokhlin, V.: An accelerated kernel-independent fast multipole method in one dimension. SIAM J. Sci. Comput. 29(3), 1160–1178 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Martinsson, P.G., Rokhlin, V.: A fast direct solver for scattering problems involving elongated structures. J. Comput. Phys. 221(1), 288–302 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010).
  38. 38.
    Sethian, J.A., Strain, J.: Crystal growth and dendritic solidification. J. Comput. Phys. 98, 231–253 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Spivak, M., Veerapaneni, S.K., Greengard, L.: The fast generalized Gauss transform. SIAM J. Sci. Comput. 32, 3092–3107 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Tausch, J.: A fast method for solving the heat equation by layer potentials. J. Comput. Phys 224(2), 956–969 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Wang, J.: Integral equation methods for the heat equation in moving geometry. Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, New York (2017)Google Scholar
  42. 42.
    Wang, J., Greengard, L., Jiang, S., Veerapaneni, S.: An efficient bootstrap method for the heat equation in moving geometry (2018) (in preparation)Google Scholar
  43. 43.
    Wang, S.: Efficient high-order integral equation methods for the heat equation. Ph.D. thesis, Department of Mathematical Sciences, New Jersey Institute of Technology, Newark (2016)Google Scholar
  44. 44.
    Ying, L., Biros, G., Zorin, D.: A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys. 196, 591–626 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

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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