Iterative schemes for high order compact discretizations to the exterior Helmholtz equation
ESAIM: Mathematical Modelling and Numerical Analysis , Special volume in honor of Professor David Gottlieb. Numéro spécial, Tome 46 (2012) no. 3, pp. 647-660.

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.

DOI : 10.1051/m2an/2011063
Classification : 35J47, 65M06
Mots-clés : Helmholtz equation, high order compact schemes 
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     title = {Iterative schemes for high order compact discretizations to the exterior {Helmholtz} equation},
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Erlangga, Yogi; Turkel, Eli. Iterative schemes for high order compact discretizations to the exterior Helmholtz equation. ESAIM: Mathematical Modelling and Numerical Analysis , Special volume in honor of Professor David Gottlieb. Numéro spécial, Tome 46 (2012) no. 3, pp. 647-660. doi : 10.1051/m2an/2011063. http://www.numdam.org/articles/10.1051/m2an/2011063/

[1] I.M. Babuška and S.A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev. 42 (2000) 451-484. | MR | Zbl

[2] A. Bayliss, C.I. Goldstein and E. Turkel, An iterative method for the Helmholtz equation. J. Comput. Phys. 49 (1983) 443-457. | MR | Zbl

[3] A. Bayliss, C.I. Goldstein and E. Turkel, On accuracy conditions for the numerical computation of waves. J. Comput. Phys. 59 (1985) 396-404. | MR | Zbl

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

[5] 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.

[6] H.C. Elman and D.P. O'Leary, Efficient iterative solution of the three dimensional Helmholtz equation. J. Comput. Phys. 142 (1998) 163-181. | Zbl

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

[8] Y.A. Erlangga, C. Vuik and C.W. Oosterlee, On a class of preconditioners for the Helmholtz equation. Appl. Numer. Math. 50 (2004) 409-425. | MR | Zbl

[9] Y.A. Erlangga, C.W. Oosterlee and C. Vuik, A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27 (2006) 1471-1492. | MR | Zbl

[10] Y.A. Erlangga, C. Vuik and C.W. Oosterlee, Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. Appl. Numer. Math. 56 (2006) 648-666. | MR | Zbl

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

[12] I. Harari and E. Turkel, Accurate finite difference methods for time-harmonic wave propagation. J. Comput. Phys. 119 (1995) 252-270. | MR | Zbl

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

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

[15] H. Tal-Ezer and E. Turkel, Iterative Solver for the Exterior Helmholtz Problem. SIAM J. Sci. Comput. 32 (2010) 463-475. | MR | Zbl

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

[17] 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).

[18] H.A. Van Der Vorst, 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 | Zbl

[19] M.B. Van Gijzen, Y.A. Erlangga and C. Vuik, Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplace precondtioner. SIAM J. Sci. Comput. 29 (2006) 1942-1958. | MR | Zbl

[20] R. Wienands, C.W. Oosterlee, On three-grid Fourier analysis for multigrid. SIAM J. Sci. Comput. 22 (2001) 651-671. | MR | Zbl

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