The Fully Discrete Legendre Spectral Approximation of Two-Dimensional Unsteady Incompressible Fluid Flow in Stream Function Form

Abstract

The initial-boundary value problem of two-dimensional incompressible fluid flow in stream function form is considered. A fully discrete Legendre spectral scheme is proposed. By a series of a priori estimations and a compactness argument, it is proved that the numerical solution converges to the weak solution of the original problem. If the genuine solution is suitably smooth, then this approach provides higher accuracy. The numerical results show the advantages of this method. The techniques used in this paper are also applicable to other related problems with derivatives of high order in space.