We consider high order finite difference approximations to the Helmholtz equation in an exterior domain. We include a simplified absorbing boundary condition to approximate the Sommerfeld radiation condition. This yields a large, but sparse, complex system, which is not self-adjoint and not positive definite. We discretize the equation with a compact fourth or sixth order accurate scheme. We solve this large system of linear equations with a Krylov subspace iterative method. Since the method converges slowly, a preconditioner is introduced, which is a Helmholtz equation but with a modified complex wavenumber. This is discretized by a second or fourth order compact scheme. The system is solved by BICGSTAB with multigrid used for the preconditioner. We study, both by Fourier analysis and computations this preconditioned system especially for the effects of high order discretizations.

Classification: 35J47, 65M06

Keywords: Helmholtz equation, high order compact schemes

@article{M2AN_2012__46_3_647_0, author = {Erlangga, Yogi and Turkel, Eli}, title = {Iterative schemes for high order compact discretizations to the exterior Helmholtz equation}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {46}, number = {3}, year = {2012}, pages = {647-660}, doi = {10.1051/m2an/2011063}, zbl = {1272.65082}, mrnumber = {2877369}, language = {en}, url = {http://www.numdam.org/item/M2AN_2012__46_3_647_0} }

Erlangga, Yogi; Turkel, Eli. Iterative schemes for high order compact discretizations to the exterior Helmholtz equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 3, pp. 647-660. doi : 10.1051/m2an/2011063. http://www.numdam.org/item/M2AN_2012__46_3_647_0/

[1] Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev. 42 (2000) 451-484. | MR 1786934 | Zbl 0956.65095

and ,[2] An iterative method for the Helmholtz equation. J. Comput. Phys. 49 (1983) 443-457. | MR 701181 | Zbl 0524.65068

, and ,[3] On accuracy conditions for the numerical computation of waves. J. Comput. Phys. 59 (1985) 396-404. | MR 794614 | Zbl 0647.65072

, and ,[4] Multi-level adaptive solution to the boundary- value problems. Math. Comp. 31 (1977) 333-390. | MR 431719 | Zbl 0373.65054

,[5] Remarks on the wave-ray Multigrid Solvers for Helmholtz Equations, Computational Fluid and Solid Mechanics, edited by K.J. Bathe. Elsevier (2003) 1871-1871.

and ,[6] Efficient iterative solution of the three dimensional Helmholtz equation. J. Comput. Phys. 142 (1998) 163-181. | Zbl 0929.65089

and ,[7] Advances in iterative methods and preconditioners for the Helmholtz equation. Arch. Comput. Methods Eng. 15 (2008) 37-66. | MR 2387515 | Zbl 1158.65078

,[8] On a class of preconditioners for the Helmholtz equation. Appl. Numer. Math. 50 (2004) 409-425. | MR 2074012 | Zbl 1051.65101

, and ,[9] A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27 (2006) 1471-1492. | MR 2199758 | Zbl 1095.65109

, and ,[10] Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. Appl. Numer. Math. 56 (2006) 648-666. | MR 2211499 | Zbl 1094.65041

, and ,[11] A complex Jacobi iterative method for the indefinite Helmholtz equation. J. Comput. Phys. 203 (2005) 358-370. | Zbl 1069.65110

,[12] Accurate finite difference methods for time-harmonic wave propagation. J. Comput. Phys. 119 (1995) 252-270. | MR 1342918 | Zbl 0848.65072

and ,[13] High order finite difference methods for the Helmholtz equation. Comput. Meth. Appl. Mech. Eng. 163 (1998) 343-358. | MR 1653767 | Zbl 0940.65112

and ,[14] Sixth order accurate finite difference schemes for the Helmholtz equation. J. Comp. Acous. 14 (2006) 339-351. | MR 2263557 | Zbl 1198.65210

and ,[15] Iterative Solver for the Exterior Helmholtz Problem. SIAM J. Sci. Comput. 32 (2010) 463-475. | MR 2599786 | Zbl 1209.65041

and ,[16] Numerical methods and nature. J. Sci. Comput. 28 (2006) 549-570. | MR 2272646 | Zbl 1158.76386

,[17] Boundary Conditions and Iterative Schemes for the Helmholtz Equation in Unbounded Regions, Computational Methods for Acoustics Problems, edited by F. Magoules. Saxe-Coburg Publ. UK (2008).

,[18] Bi-CGSTAB : A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13 (1992) 631-644. | MR 1149111 | Zbl 0761.65023

,[19] Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplace precondtioner. SIAM J. Sci. Comput. 29 (2006) 1942-1958. | MR 2350014 | Zbl 1155.65088

, and ,[20] On three-grid Fourier analysis for multigrid. SIAM J. Sci. Comput. 22 (2001) 651-671. | MR 1861270 | Zbl 0992.65137

, ,