Chebyshev pseudospectral method for wave equation with absorbing boundary conditions that does not use a first order hyperbolic system

The analysis and solution of wave equations with absorbing boundary conditions by using a related first order hyperbolic system has become increasingly popular in recent years. At variance with several methods which rely on this transformation, we propose an alternative method in which such hyperbolic system is not used. The method consists of approximation of spatial derivatives by the Chebyshev pseudospectral collocation method coupled with integration in time by the Runge-Kutta method. Stability limits on the timestep for arbitrary speed are calculated and verified numerically. Furthermore, theoretical properties of two methods by Jackiewicz and Renaut are derived, including, in particular, a result that corrects some conclusions of these authors. Numerical results that verify the theory and illustrate the effectiveness of the proposed approach are reported.