Direct Solution of the Vorticity-Stream Function Ordinary Differential Equations by a Chebyshev Approximation

A numerical method for solving the coupled vorticity-stream function equations in one dimension with an exact noniterative determination of the vorticity boundary values is presented. The one-dimensional form considered can represent the Fourier modes of a two-dimensional problem. The equations are separated by replacing the derivative specifications for the stream function at the boundary points with equivalent conditions of an integral type for the vorticity. A spectral approximation by means of Chebyshev polynomials is considered. The numerical properties of the algorithm are investigated against a few analytical examples which demonstrate the accuracy of the proposed method.