Spectral Methods for Initial Boundary Value Problems—An Alternative Approach

Abstract

This paper presents a new approach to spectral methods for initial boundary value problems. A filtered version of the partial differential equation and the initial and boundary conditions at an overdetermined set of points are collocated. As an approximate solution, the function is chosen that belongs to an appropriate finite-dimensional space and minimizes a weighted average of the residuals at these points. It is proved that the approximate solution converges to the actual solution at a spectral rate of accuracy in both space and time. The proof is based on a priori energy estimates that have been proved for such systems. Although this method is restricted here to hyperbolic initial boundary value problems and Chebyshev polynomials, it generalizes to general initial boundary value problems, boundary value problems, and Gegenbauer polynomials.