Stability of Gauss-Radau Pseudospectral Approximations of the One-Dimensional Wave Equation

We first extend the stability analysis. of pseudospectral approximations of the one-dimensional one-way wave equation ∂u/∂x = c(x) ∂u/∂x given in [11] to general Gauss-Radau collocation methods. We give asufficient condition on the collocation points for stability whichshows that classical Gauss-Radau ultraspherical methods are perfectly stable while their Gauss-Lobatto counterpart is not. When the stability condition is not met we introduce a simple modification of the approximation which leads to better stability properties. Numerical examples show that long term stability may substantially improve.