Analysis of Chebyshev Collocation Methods for Parabolic Equations

Abstract

Time discretizations of both the heat equation and the advection-diffusion equation in two space variables are analyzed. For space approximation the pseudospectral Chebyshev method, which enjoys infinite order of accuracy for smooth solutions, is used. The equation is collocated at the Gauss–Chebyshev points, and derivatives are computed by the pseudospectral differencing technique. At any time interval the pseudospectral solution, which is a polynomial of degree N, is advanced in time using the implicit $\theta $-method for the diffusive part of the equation, while the advective term is dealt with explicitly. We prove unconditional stability and optimal error bounds, depending on both N and $\Delta t$ (the time-step), in the norms of the weighted Sobolev spaces. The method considered here is the most commonly used spectral method for parabolic equations. To solve the linear system arising at any time interval, efficient iterative techniques with scaling are presented.