Journal of Scientific Computing

, Volume 78, Issue 3, pp 1632–1658

# A Fourth-Order Kernel-Free Boundary Integral Method for the Modified Helmholtz Equation

• Yaning Xie
• Wenjun Ying
Article

## Abstract

Based on the kernel-free boundary integral method proposed by Ying and Henriquez (J Comput Phys 227(2):1046–1074, 2007), which is a second-order accurate method for general elliptic partial differential equations, this work develops it to be a fourth-order accurate version for the modified Helmholtz equation. The updated method is in line with the original one. Unlike the traditional boundary integral method, it does not need to know any analytical expression of the fundamental solution or Green’s function in evaluation of boundary or volume integrals. Boundary value problems under consideration are reformulated into Fredholm boundary integral equations of the second kind, whose corresponding discrete forms are solved with the simplest Krylov subspace iterative method, the Richardson iteration. During each iteration, a Cartesian grid based nine-point compact difference scheme is used to discretize the simple interface problem whose solution is the boundary or volume integral in the BIEs. The resulting linear system is solved by a fast Fourier transform based solver, whose computational work is roughly proportional to the number of grid nodes in the Cartesian grid used. As the discrete boundary integral equations are well-conditioned, the iteration converges within an essentially fixed number of steps, independent of the mesh parameter. Numerical results are presented to verify the solution accuracy and demonstrate the algorithm efficiency.

## Keywords

Elliptic partial differential equation Kernel-free boundary integral method Cartesian grid method Nine-point compact difference scheme

## References

1. 1.
Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)
2. 2.
Ben Avraham, D., Fokas, A.S.: The solution of the modified Helmholtz equation in a wedge and an application to diffusion-limited coalescence. Phys. Lett. A 263(4–6), 355–359 (1999)
3. 3.
Bakker, M.: Modeling groundwater flow to elliptical lakes and through multi-aquifer elliptical inhomogeneities. Adv. Water Resour. 27(5), 497–506 (2004)Google Scholar
4. 4.
Bakker, M., Kuhlman, K.L.: Computational issues and applications of line-elements to model subsurface flow governed by the modified Helmholtz equation. Adv. Water Resour. 34(9), 1186–1194 (2011)Google Scholar
5. 5.
Banerjee, P.K., Butterfield, R.: Boundary Element Methods in Engineering Science, vol. 17. McGraw-Hill, London (1981)
6. 6.
Barnes, J., Hut, P.: A hierarchical O(N log N) force-calculation algorithm. Nature 324(6096), 446 (1986)Google Scholar
7. 7.
Barnes, J.E.: A modified tree code: Don’t laugh. It runs. J. Comput. Phys. 87(1), 161–170 (1990)
8. 8.
Bazant, M.Z., Thornton, K., Ajdari, A.: Diffuse-charge dynamics in electrochemical systems. Phys. Rev. E 70(2), 021506 (2004)Google Scholar
9. 9.
Beale, J.T., Layton, A.T.: On the accuracy of finite difference methods for elliptic problems with interfaces. Commun. Appl. Math. Comput. Sci. 1(1), 91–119 (2006)
10. 10.
Brebbia, C., Dominguez, J.: Boundary element methods for potential problems. Appl. Math. Model. 1(7), 372–378 (1977)
11. 11.
Chapko, R., Kress, R.: Rothe’s method for the heat equation and boundary integral equations. J. Integral Equ. Appl. 9, 47–69 (1997)
12. 12.
Cheng, H., Crutchfield, W.Y., Gimbutas, Z., Greengard, L.F., Ethridge, J.F., Huang, J., Rokhlin, V., Yarvin, N., Zhao, J.: A wideband fast multipole method for the Helmholtz equation in three dimensions. J. Comput. Phys. 216(1), 300–325 (2006)
13. 13.
Cheng, H., Greengard, L., Rokhlin, V.: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155(2), 468–498 (1999)
14. 14.
Cheng, H., Huang, J., Leiterman, T.J.: An adaptive fast solver for the modified Helmholtz equation in two dimensions. J. Comput. Phys. 211(2), 616–637 (2006)
15. 15.
Di Gioia, A.: Fast multipole accelerated boundary element techniques for large-scale problems, with applications to MEMS. Ph.D. thesis, Università di Trento. Dipartimento di ingegneria meccanica e strutturale (2005)Google Scholar
16. 16.
Fedkiw, R.P., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152(2), 457–492 (1999)
17. 17.
Feng, H., Barua, A., Li, S., Li, X.: A parallel adaptive treecode algorithm for evolution of elastically stressed solids. Commun. Comput. Phys. 15(2), 365–387 (2014)
18. 18.
Gibou, F., Fedkiw, R., Cheng, L.T., Kang, M.: A second order accurate symmetric discretization of the Poisson equation on irregular domains. J. Comput. Phys. 176, 205–227 (2002)
19. 19.
Gibou, F., Fedkiw, R.P.: A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem. J. Comput. Phys. 202, 577–601 (2005)
20. 20.
Greengard, L., Huang, J., Rokhlin, V., Wandzura, S.: Accelerating fast multipole methods for the Helmholtz equation at low frequencies. IEEE Comput. Sci. Eng. 5(3), 32–38 (1998)Google Scholar
21. 21.
Greengard, L., Kropinski, M.C.: An integral equation approach to the incompressible Navier–Stokes equations in two dimensions. SIAM J. Sci. Comput. 20(1), 318–336 (1998)
22. 22.
Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)
23. 23.
Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numer. 6, 229–269 (1997)
24. 24.
Hackbusch, W.: Integral Equations, Theory and Numerical Treatment. Birkhäuser, Basel (1995)
25. 25.
Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54(4), 463–491 (1989)
26. 26.
He, X., Lin, T., Lin, Y.: Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions. Int. J. Numer. Anal. Model. 8(2), 284–301 (2011)
27. 27.
Hewett, D.W.: The embedded curved boundary method for orthogonal simulation meshes. J. Comput. Phys. 138(2), 585–616 (1997)
28. 28.
Hou, S., Liu, X.D.: A numerical method for solving variable coefficient elliptic equation with interfaces. J. Comput. Phys. 202, 411–445 (2005)
29. 29.
Houstis, E.N., Papatheodorou, T.S.: Algorithm 543: FFT9, fast solution of Helmholtz-type partial differential equations D3. ACM Trans. Math. Softw. 5(4), 490–493 (1979)
30. 30.
Houstis, E.N., Papatheodorou, T.S.: High-order fast elliptic equation solver. ACM Trans. Math. Softw. 5(4), 431–441 (1979)
31. 31.
Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer, Berlin (2008)
32. 32.
Jaswon, M.: Integral equation methods in potential theory. I. In: Proceedings of the Royal Society of London A, vol. 275, pp. 23–32. The Royal Society (1963)Google Scholar
33. 33.
Johansen, H., Colella, P.: A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains. J. Comput. Phys. 147(1), 60–85 (1998)
34. 34.
Kropinski, M.C.A., Quaife, B.D.: Fast integral equation methods for Rothe’s method applied to the isotropic heat equation. Comput. Math. Appl. 61(9), 2436–2446 (2011)
35. 35.
Kropinski, M.C.A., Quaife, B.D.: Fast integral equation methods for the modified Helmholtz equation. J. Comput. Phys. 230(2), 425–434 (2011)
36. 36.
Kuhlman, K.L., Neuman, S.P.: Laplace-transform analytic-element method for transient porous-media flow. J. Eng. Math. 64(2), 113 (2009)
37. 37.
Le, D.V., Khoo, B.C., Peraire, J.: An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries. J. Comput. Phys. 220(1), 109–138 (2006)
38. 38.
Leveque, R.J., Li, Z.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31(4), 1019–1044 (1994)
39. 39.
Li, H., Huang, J.: High accuracy solutions of the modified Helmholtz equation. In: Information Technology and Intelligent Transportation Systems, pp. 29–37. Springer (2017)Google Scholar
40. 40.
Li, Z.: A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35(1), 230–254 (1998)
41. 41.
Li, Z., Ito, K.: The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, vol. 33. SIAM, Philadelphia (2006)
42. 42.
Lindsay, K., Krasny, R.A.: A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow. J. Comput. Phys. 172(2), 879–907 (2001)
43. 43.
Liu, Y., Nishimura, N.: The fast multipole boundary element method for potential problems: a tutorial. Eng. Anal. Bound. Elem. 30(5), 371–381 (2006)
44. 44.
Marques, A.N., Nave, J.C., Rosales, R.R.: A correction function method for Poisson problems with interface jump conditions. J. Comput. Phys. 230, 7567–7597 (2011)
45. 45.
Marques, A.N., Nave, J.C., Rosales, R.R.: High order solution of Poisson problems with piecewise constant coefficients and interface jumps. J. Comput. Phys. 335, 497–515 (2017)
46. 46.
Mayo, A.: The fast solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J. Numer. Anal. 21(2), 285–299 (1984)
47. 47.
Mayo, A.: Fast high order accurate solution of Laplace’s equation on irregular regions. SIAM J. Sci. Stat. Comput. 6(1), 144–157 (1985)
48. 48.
Mayo, A.: The rapid evaluation of volume integrals of potential theory on general regions. J. Comput. Phys. 100(2), 236–245 (1992)
49. 49.
Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)
50. 50.
Phillips, J.R., White, J.K.: A precorrected-FFT method for electrostatic analysis of complicated 3-D structures. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 16(10), 1059–1072 (1997)Google Scholar
51. 51.
Politis, C.G., Papalexandris, M.V., Athanassoulis, G.A.: A boundary integral equation method for oblique water-wave scattering by cylinders governed by the modified helmholtz equation. Appl. Ocean Res. 24(4), 215–233 (2002)Google Scholar
52. 52.
Quaife, B.D.: Fast integral equation methods for the modified Helmholtz equation. Ph.D. thesis, Science: Department of Mathematics (2011)Google Scholar
53. 53.
Rokhlin, V.: Rapid solution of integral equations of classical potential theory. J. Comput. Phys. 60(2), 187–207 (1985)
54. 54.
Rosser, J.B.: Nine-point difference solutions for Poisson’s equation. Comput. Math. Appl. 1(3–4), 351–360 (1975)
55. 55.
Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)
56. 56.
Samarskii, A.A.: The Theory of Difference Schemes, vol. 240. CRC Press, London (2001)
57. 57.
Sauter, S., Schwab, C.: Boundary Element Methods. Springer, Berlin (2010)
58. 58.
Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer, Berlin (2007)Google Scholar
59. 59.
Strack, O.D.: Groundwater Mechanics. Prentice Hall, Englewood Cliffs (1989)Google Scholar
60. 60.
Ying, L., Biros, G., Zorin, D.: A kernel-independent adaptive fast multipole algorithm in two and three space dimensions. J. Comput. Phys. 196, 591–626 (2004)
61. 61.
Ying, L., Biros, G., Zorin, D.: A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains. J. Comput. Phys. 219, 247–275 (2006)
62. 62.
Ying, W., Henriquez, C.S.: A kernel-free boundary integral method for elliptic boundary value problems. J. Comput. Phys. 227(2), 1046–1074 (2007)
63. 63.
Ying, W., Wang, W.C.: A kernel-free boundary integral method for implicitly defined surfaces. J. Comput. Phys. 252, 606–624 (2013)
64. 64.
Ying, W., Wang, W.C.: A kernel-free boundary integral method for variable coefficients elliptic PDEs. Commun. Comput. Phys. 15(4), 1108–1140 (2014)
65. 65.
Zhou, Y.C., Zhao, S., Feig, M., Wei, G.W.: High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys. 213(1), 1–30 (2006)