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

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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.

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References

  1. Alpert, B.K.: Hybrid Gauss-trapezoidal quadrature rules. SIAM J. Sci. Comput. 20(5), 1551–1584 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barnett, A., Magland, J.: Non-uniform fast Fourier transform library of types 1, 2, 3 in dimensions 1, 2, 3. https://github.com/ahbarnett/finufft (2018)

  3. Brattkus, K., Meiron, D.I.: Numerical simulations of unsteady crystal growth. SIAM J. Appl. Math. 52, 1303–1320 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brown, M.: The method of layer potentials for the heat equation in Lipschitz cylinders. Am. J. Math. 111, 339–379 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brown, M.: The initial-Neumann problem for the heat equation in Lipschitz cylinders. Trans. Am. Math. Soc. 320, 1–52 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cheng, H., Greengard, L., Rokhlin, V.: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155(2), 468–498 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  10. Fong, W., Darve, E.: The black-box fast multipole method. J. Comput. Phys. 228(23), 8712–8725 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gimbutas, Z., Rokhlin, V.: A generalized fast multipole method for nonoscillatory kernels. SIAM J. Sci. Comput. 24, 796–817 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Greengard, L., Lee, J.: Accelerating the nonuniform fast Fourier transform. SIAM Rev. 46, 443–454 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Greengard, L., Lin, P.: Spectral approximation of the free-space heat kernel. Appl. Comput. Harmon. Anal. 9, 83–97 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Greengard, L., Strain, J.: A fast algorithm for the evaluation of heat potentials. Commun. Pure Appl. Math. 43, 949–963 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Greengard, L., Strain, J.: The fast Gauss transform. SIAM J. Sci. Statist. Comput. 12, 79–94 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Greengard, L., Sun, X.: A new version of the fast Gauss transform. Doc. Math. III, 575–584 (1990)

    MathSciNet  MATH  Google Scholar 

  19. Guenther, R.B., Lee, J.W.: Partial Differential Equations of Mathematical Physics and Integral Equations. Prentice-Hall, Englewood Cliffs (1988)

    Google Scholar 

  20. Hackbusch, W., Börm, S.: Data-sparse approximation by adaptive H2-matrices. Computing 69(1), 1–35 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ho, K.L., Ying, L.: Hierarchical interpolative factorization for elliptic operators: integral equations. Commun. Pure Appl. Math. 69(7), 1314–1353 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kress, R.: Linear integral equations. In: Applied Mathematical Sciences, vol. 82, 3rd edn. Springer, Berlin (2014)

  30. Lee, J.Y., Greengard, L., Gimbutas, Z.: NUFFT Version 1.3.2 Software Release. http://www.cims.nyu.edu/cmcl/nufft/nufft.html (2009)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, J., Greengard, L.: High order accurate methods for the evaluation of layer heat potentials. SIAM J. Sci. Comput. 31, 3847–3860 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Martinsson, P.G.: A fast direct solver for a class of elliptic partial differential equations. J. Sci. Comput. 38(3), 316–330 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  36. Martinsson, P.G., Rokhlin, V.: A fast direct solver for scattering problems involving elongated structures. J. Comput. Phys. 221(1), 288–302 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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). http://dlmf.nist.gov

  38. Sethian, J.A., Strain, J.: Crystal growth and dendritic solidification. J. Comput. Phys. 98, 231–253 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  39. Spivak, M., Veerapaneni, S.K., Greengard, L.: The fast generalized Gauss transform. SIAM J. Sci. Comput. 32, 3092–3107 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Tausch, J.: A fast method for solving the heat equation by layer potentials. J. Comput. Phys 224(2), 956–969 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  42. Wang, J., Greengard, L., Jiang, S., Veerapaneni, S.: An efficient bootstrap method for the heat equation in moving geometry (2018) (in preparation)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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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Authors and Affiliations

  1. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 07102, USA

    Shaobo Wang, Shidong Jiang & Jing Wang

  2. School of Information Science and Engineering, Yunnan University, Kunming, Yunnan, 650091, China

    Jing Wang

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  1. Shaobo Wang
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Correspondence to Jing Wang.

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Wang, S., Jiang, S. & Wang, J. Fast High-Order Integral Equation Methods for Solving Boundary Value Problems of Two Dimensional Heat Equation in Complex Geometry. J Sci Comput 79, 787–808 (2019). https://doi.org/10.1007/s10915-018-0872-x

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  • DOI: https://doi.org/10.1007/s10915-018-0872-x

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