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://archive.numdam.org/articles/10.1051/m2an/2011063/

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