Optimized finite difference coefficients for the Helmholtz equation

Abstract

Optimized finite difference (OFD) coefficients are often used to minimize numerical dispersion and to improve accuracy in finite difference (FD) solutions to partial differential equations (Lele (1992); Tam and Webb (1993); Jo et al. (1996); Liu and Sen (2010); Štekl and Pratt (1998)). We present a framework for deriving such coefficients which at once minimizes numerical dispersion and preserves convergence at low frequency. We compute optimal coefficients in each dimension and then assemble the optimal multi-dimensional stencil for the computational grid. We demonstrate the effectiveness of our OFD scheme by computing solutions to the Helmholtz equation below 3.5 points-per-wavelength (ppw).