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

Yogi Erlangga; Eli Turkel

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 3, page 647-660
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Erlangga, Yogi, and Turkel, Eli. "Iterative schemes for high order compact discretizations to the exterior Helmholtz equation∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.3 (2012): 647-660. <http://eudml.org/doc/276380>.

@article{Erlangga2012,
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.},
author = {Erlangga, Yogi, Turkel, Eli},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Helmholtz equation; high order compact schemes; convergence; numerical examples; finite difference approximations; exterior domain; Sommerfeld radiation condition; Krylov subspace iterative method; BICGSTAB; multigrid; preconditioner},
language = {eng},
month = {1},
number = {3},
pages = {647-660},
publisher = {EDP Sciences},
title = {Iterative schemes for high order compact discretizations to the exterior Helmholtz equation∗},
url = {http://eudml.org/doc/276380},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Erlangga, Yogi
AU - Turkel, Eli
TI - Iterative schemes for high order compact discretizations to the exterior Helmholtz equation∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/1//
PB - EDP Sciences
VL - 46
IS - 3
SP - 647
EP - 660
AB - 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.
LA - eng
KW - Helmholtz equation; high order compact schemes; convergence; numerical examples; finite difference approximations; exterior domain; Sommerfeld radiation condition; Krylov subspace iterative method; BICGSTAB; multigrid; preconditioner
UR - http://eudml.org/doc/276380
ER -

References

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