Spectrally accurate numerical solution of acoustic wave equations

Abstract

Finite difference models of wave propagation have presented challenging problems of stability and accuracy since initial experimentation with these models began on early digital computers. The advent of spectral methods in the late 1960s has led to the latter's increasing use for solving differential equations in a range of fluid dynamic, electromagnetic and thermal applications. Spectral methods transform a physical grid of state variables (such as acoustic velocity and pressure) into an alternative spectral space characterized by a particular set of basis functions. Spatial derivatives of physical state variables are computed in spectral space using exact differential operators expressed in terms of those functions. Fast numerical transforms are employed to exchange immediate state of a simulation between its spectral and physical representations. In problems equally suited to spectral and finite difference formulation, spectral methods often yield increased fidelity of physical results and improved stability. Spectral methods sometimes enable computational grid size requirements of a simulation to be substantially reduced, with concomitant computational savings. This paper reports on spectral implementations of the acoustic wave equation and Webster horn equation for simulating audio transducer cavities, musical instrument resonators, and the human vocal tract.