The spectrum of a Chebyshev-Fourier approximation for the Stokes equations

Abstract

A Chebyshev-Fourier approximation to the solution of the two-dimensional Stokes equations in the vorticity-streamfunction formulation is considered. The expansion in a Fourier series in the direction of periodicity leads to a family of one-dimensional Stokes-type problems being approximated by a Chebyshevcollocation method. First some results about the spectrum of the corresponding operators are derived. Then we consider a discretization in time by means of a class of semi-implicit finite differences schemes and we describe the influence matrix technique used to solve the resulting system at every time step. The properties of the spectrum of the Chebyshev-Stokes operator are used to derive some results about the stability of the resulting time marching algorithm.