Stability of a Chebychev pseudospectral solution of the wave equation with absorbing boundaries

Stability of the pseudospectral Chebychev collocation solution of the two-dimensional acoustic wave problem with absorbing boundary conditions is investigated. The continuous one-dimensional problem with one absorbing boundary and one Dirichlet boundary has previously been shown to be far from normal. Consequently, the spectrum of that problem says little about the stability behavior of the solution. Our analysis proves that the discrete formulation with Dirichlet boundaries at all boundaries is near normal and hence the formulation with absorbing boundaries at all boundaries, either for one-dimensional or two-dimensional wave propagation, is not far from normal. The near-normality follows from the near-normality of the second-order derivative pseudospectral differential operator. Further, the nearness to normality is independent of the boundary discretization. Stability limits on the timestep are, however, dependent on the boundary operator, with an explicit Euler method having the most restrictive condition. The Crank-Nicolson implementation has a stability limit the same as the Dirichlet formulation. Furthermore, in this case the restriction scales by 12 in moving from one dimension to two dimensions, exactly as in the central finite difference approximation. Numerical results confirm the predicted values on allowable timesteps obtained from a spectral analysis, for both Chebychev- and modified-Chebychev-implementations. We conclude that the spectrum of the evolution operator is informative for predicting the behavior of the numerical solution.