Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number

Eli Turkel, Dan Gordon, Rachel Gordon, Semyon Tsynkov

Research output: Contribution to journalArticlepeer-review


Several studies have presented compact fourth order accurate finite difference approximation for the Helmholtz equation in two or three dimensions. Several of these formulae allow for the wave number k to be variable. Other papers have extended this further to include variable coefficients within the Laplacian which models non-homogeneous materials in electromagnetism.Later papers considered more accurate compact sixth order methods but these were restricted to constant k. In this paper we extend these compact sixth order schemes to variable k in both two and three dimensions. Results on 2D and 3D problems with known analytic solutions verify the sixth order accuracy. We demonstrate that for large wave numbers, the second order scheme cannot produce comparable results with reasonable grid sizes.

Original languageEnglish
Pages (from-to)272-287
Number of pages16
JournalJournal of Computational Physics
Issue number1
StatePublished - 1 Jan 2013

Bibliographical note

Funding Information:
The authors thank the anonymous reviewers for their useful comments. The research of the first and fourth authors was partially supported by the US-Israel Binational Science Foundation (BSF) under Grant No. 2008094 . The work of the fourth author was also partially supported by the US ARO under Grant No. W911NF-11-1-0384 .


  • Compact high order schemes
  • Helmholtz equation
  • High frequency
  • Large wave number
  • Parallel computing
  • Variable wave number

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy (all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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