A note on stability of pseudospectral methods for wave propagation

In this paper we deal with the effects on stability of subtle differences in formulations of pseudospectral methods for solution of the acoustic wave equation. We suppose that spatial derivatives are approximated by Chebyshev pseudospectral discretizations. Through reformulation of the equations as first order hyperbolic systems any appropriate ordinary differential equation solver can be used to integrate in time. However, the resulting stability, and hence efficiency, properties of the numerical algorithms are drastically impacted by the manner in which the absorbing boundary conditions are incorporated. Specifically, mathematically equivalent well-posed approaches are not equivalent numerically. An analysis of the spectrum of the resultant system operator predicts these properties.