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

Authors:
Eli Turkel;Dan Gordon;Rachel Gordon;Semyon Tsynkov
Affiliations:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel;Department of Computer Science, University of Haifa, Haifa 31905, Israel;Department of Aerospace Engineering, The Technion - Israel Institute of Technology, Haifa 32000, Israel;Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
Venue:
Journal of Computational Physics
Year:
2013

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Abstract

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.