A high-order accurate solution method for complex geometries is developed for two-dimensional flows using the stream function-vorticity formulation. High-order accurate spectrally optimized compact schemes along with appropriate boundary schemes are used for spatial discretization while a two-level backward Euler implicit scheme is used for the time integration. The linear system of equations for stream function and vorticity are solved by an inner iteration while contravariant velocities constitute outer iterations. The effect of curvilinear grids on the solution accuracy is studied. The method is used to compute Cartesian and inclined driven cavity, flow in a triangular cavity and viscous flow in constricted channel. Benchmark-like accuracy is obtained in all the problems with fewer grid points compared to reported studies. Copyright © 2006 John Wiley & Sons, Ltd.