Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-12-01T06:55:41.037Z Has data issue: false hasContentIssue false
  • English
  • Français

Iterative schemes for high order compact discretizations to the exterior Helmholtz equation∗

Published online by Cambridge University Press:  11 January 2012

Yogi Erlangga  and
Eli Turkel
Yogi Erlangga
Affiliation:
Department of Earth and Ocean Sciences, The University of British Columbia, 2329 West Mall Vancouver, BC V6T 124, Canada Currently at Mathematics Division, College of Sciences, Alfaisal University, P.O. Box 50927, Riyadh, 11533 Kingdom of Saudi Arabia
Eli Turkel
Affiliation:
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel. [email protected]
Get access

Abstract

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.

Keywords

Helmholtz equationhigh order compact schemes
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 3: Special volume in honor of Professor David Gottlieb , May 2012 , pp. 647 - 660
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Babuška, I.M. and Sauter, S.A., Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev. 42 (2000) 451–484. Google Scholar
Bayliss, A., Goldstein, C.I. and Turkel, E., An iterative method for the Helmholtz equation. J. Comput. Phys. 49 (1983) 443–457. Google Scholar
Bayliss, A., Goldstein, C.I. and Turkel, E., On accuracy conditions for the numerical computation of waves. J. Comput. Phys. 59 (1985) 396–404. Google Scholar
Brandt, A., Multi-level adaptive solution to the boundary- value problems. Math. Comp. 31 (1977) 333-390. Google Scholar
A. Brandt and I. Livshits, Remarks on the wave-ray Multigrid Solvers for Helmholtz Equations, Computational Fluid and Solid Mechanics, edited by K.J. Bathe. Elsevier (2003) 1871–1871.
Elman, H.C. and O’Leary, D.P., Efficient iterative solution of the three dimensional Helmholtz equation. J. Comput. Phys. 142 (1998) 163–181. Google Scholar
Erlangga, Y.A., Advances in iterative methods and preconditioners for the Helmholtz equation. Arch. Comput. Methods Eng. 15 (2008) 37–66. Google Scholar
Erlangga, Y.A., Vuik, C. and Oosterlee, C.W., On a class of preconditioners for the Helmholtz equation. Appl. Numer. Math. 50 (2004) 409–425. Google Scholar
Erlangga, Y.A., Oosterlee, C.W. and Vuik, C., A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27 (2006) 1471–1492. Google Scholar
Erlangga, Y.A., Vuik, C. and Oosterlee, C.W., Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. Appl. Numer. Math. 56 (2006) 648–666. Google Scholar
Hadley, G.R., A complex Jacobi iterative method for the indefinite Helmholtz equation. J. Comput. Phys. 203 (2005) 358–370. Google Scholar
Harari, I. and Turkel, E., Accurate finite difference methods for time-harmonic wave propagation. J. Comput. Phys. 119 (1995) 252–270. Google Scholar
Singer, I. and Turkel, E., High order finite difference methods for the Helmholtz equation. Comput. Meth. Appl. Mech. Eng. 163 (1998) 343–358. Google Scholar
Singer, I. and Turkel, E., Sixth order accurate finite difference schemes for the Helmholtz equation. J. Comp. Acous. 14 (2006) 339–351. Google Scholar
Tal-Ezer, H. and Turkel, E., Iterative Solver for the Exterior Helmholtz Problem. SIAM J. Sci. Comput. 32 (2010) 463–475. Google Scholar
Turkel, E., Numerical methods and nature. J. Sci. Comput. 28 (2006) 549–570. Google Scholar
E. Turkel, 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).
van der Vorst, H.A., 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. Google Scholar
van Gijzen, M.B., Erlangga, Y.A. and Vuik, C., Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplace precondtioner. SIAM J. Sci. Comput. 29 (2006) 1942–1958. Google Scholar
Wienands, R., Oosterlee, C.W., On three-grid Fourier analysis for multigrid. SIAM J. Sci. Comput. 22 (2001) 651–671.Google Scholar