Stability of pseudospectral approximations of the one-dimensional wave equation

B. D. Welfert, Z. Jackiewicz

Department of Mathematics & Statistics, Arizona State University, Tempe, Arizona 85287, USA

bdw@asu.edu

Contributed talk

We investigate the stability of pseudospectral approximations for the one-dimensional one-way wave equation

$$\frac{\partial u}{\partial x}=c(x) \frac{\partial u}{\partial x}$$

under regular Dirichlet boundary condition $u(1,t)=g(t)$. In particular:

• we extend (asymptotic) stability results presented in [1] to general collocation methods based on a class of admissible orthogonal polynomials;
• we introduce a simple stabilization procedure which yields substantially better asymptotic properties;
• we verify numerically for several choices of $c(x)$ the stability of Jacobi methods and present the substantial improvement in long-term behavior of the solution resulting from the stabilization procedure;
• we introduce the concept of matrix filter and sketch the extension of the above ideas to higher dimensions.

Details are included in the preprint [2] available at

http://math.asu.edu/~bdw/PAPERS/stability_1D_wave.pdf.