Construction and optimization techniques for high order schemes for the two-dimensional wave equation

Abstract

With the advent of high performance parallel computing, audio rate room auralization using finite difference time domain methods is becoming possible in a reasonable computation time. Yet, there are still deficiencies in the methods which are used for this purpose, particularly with regard to minimizing numerical dispersion over the full range of audible frequencies. This paper is concerned with construction techniques for families of methods for the test case of the 2D wave equation. Such methods are explicit, can be of very high accuracy, and operate over a small local stencil. Such schemes can be attractive in a parallel computation environment. As such methods will depend, invariably, on a set of free parameters, including the Courant number, a major concern is optimization. The remainder of this paper approaches the problem of setting up such an optimization problem in terms of various constraints and a suitable cost function. Some of the constraints follow from consistency, stability, isotropy and accuracy of the resulting scheme, and others from perceptual considerations peculiar to audio. Simulation results will be presented.