Chebyshev spectral and pseudo-spectral methods in unbounded domains.

Abstract

Chebyshev type spectral methods are widely used in numerical simulations of PDEs posed in unbounded domains. Such methods have a number of important computational advantages. In particular, they admit very efficient practical implementation. However, the stability and convergence analysis of these methods require deep understanding of approximation properties of the underlying functional basis. In this project, we deal with Chebyshev spectral and pseudo-spectral methods in unbounded domains. The first part of the project deals with theoretical analysis of Chebyshev-type spectral projection and interpolation operators in Bessel potential spaces. In the second part, we provide rigorous analyses of Chebyshev-type pseudo-spectral (collocation) scheme applied to the nonlinear Schrodinger equation. The project is concluded with several numerical experiments.