Optimised 25-point finite difference schemes for the three-dimensional wave equation

Wave-based methods are increasingly viewed as necessary alternatives to geometric methods for room acoustics simulations, as they naturally capture wave phenomena like diffraction and interference. For methods that simulate the three-dimensional wave equation—and thus solve for the entire acoustic field in an enclosed space—computational costs can be high, so efficient algorithms are critical. In terms of computational complexity, finite difference schemes are possibly the simplest such algorithms, but they are known to suffer from numerical dispersion. High-order and optimised schemes can offer improved numerical dispersion, and thus, computationally efficient numerical solutions. In this paper, we consider two families of explicit finite difference schemes for the second-order wave equation in three spatial dimensions, using 25-point stencils on the Cartesian grid. We review known special cases that lead to high-order accuracy in space (and possibly in time), and we present new schemes with optimised stencil coefficients. These schemes provide accurate wave simulation using substantially less memory than the conventional scheme. Simulations are presented to demonstrate the performance of the optimised schemes.