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

Authors:
Z. Jackiewicz;B. D. Welfert
Affiliations:
Department of Mathematics, Arizona State University, Tempe, Arizona 85287. jackiewi@asu.edu;Department of Mathematics, Arizona State University, Tempe, Arizona 85287. bdw@asu.edu
Venue:
Journal of Scientific Computing
Year:
2003

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Abstract

We first extend the stability analysis of pseudospectral approximations of the one-dimensional one-way wave equation \frac{\partial u}{\partial x}=c(x) \frac {\partial u}{\partial 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.