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

Yogi Erlangga, Eli Turkel

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)647-660
Number of pages14
JournalESAIM: Mathematical Modelling and Numerical Analysis
Issue number3
StatePublished - May 2012


  • Helmholtz equation
  • High order compact schemes


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