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In this paper, a modified central finite difference scheme for a three-point nonuniform grid is presented for the one-dimensional homogeneous Helmholtz equation using the Bloch wave property. The modified scheme provides highly accurate solutions at the nodes of the nonuniform grid for very small to very large range of wave numbers irrespective of how the grid is adapted throughout the domain. | Turk J Math (2016) 40: 806 – 815 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1506-24 Research Article Simulations of the Helmholtz equation at any wave number for adaptive grids using a modified central finite difference scheme Hafiz Abdul WAJID1,2,∗ Department of Electrical Engineering, Faculty of Engineering, Islamic University in Medina, Al-Medina Al-Munawarah, Saudi Arabia 2 Department of Mathematics, Comsats Institute of Information Technology, Lahore, Pakistan 1 Received: 08.06.2015 • Accepted/Published Online: 27.10.2015 • Final Version: 16.06.2016 Abstract: In this paper, a modified central finite difference scheme for a three-point nonuniform grid is presented for the one-dimensional homogeneous Helmholtz equation using the Bloch wave property. The modified scheme provides highly accurate solutions at the nodes of the nonuniform grid for very small to very large range of wave numbers irrespective of how the grid is adapted throughout the domain. A variety of numerical examples are considered to validate the superiority of the modified scheme for a nonuniform grid over a standard central finite difference scheme. Key words: Helmholtz equation, modified central finite difference scheme, numerical dispersion, nonuniform grids, adaptive grids 1. Introduction Numerical dispersion has always been a hindrance for the accurate simulations of the Helmholtz equation and to reduce or eliminate it one needs to use extremely refined mesh, resulting in prohibitive computational cost. Reducing numerical dispersion really becomes untractable (a) in the case of large wave numbers because of the highly oscillatory behavior of waves; (b) when the problem is posed in higher dimensions because of anisotropy; (c) when the discretization of the domain is not uniform. Many efforts have already been made in the last few decades to solve the Helmholtz equation [1–4, 9, 10, 19] and it is not surprising that finite .
