Spectral methods for initial boundary value problems with nonsmooth data

In this paper we consider hyperbolic initial boundary value problems with n onsmooth data. We show that if we extend the time domain to minus infinity, replace the initial condition by a growth condition at minus infinity and then solve the problem using a filtered version of the data by the Galerkin -Collocation method using Laguerre polynomials in time and Legendre polynom ials in space, then we can recover pointwise values with spectral accuracy, provided that the actual solution is piecewise smooth. For this we have to perform a local smoothing of the computed solution.