A rational high-order compact difference method for the steady-state stream function–vorticity formulation of the Navier–Stokes equations

A rational high-order compact (RHOC) finite difference (FD) method on the nine-point stencil is proposed for solving the steady-state two-dimensional NavierStokes equations in the stream functionvorticity form. The resulting system of algebra equations can be solved by using the point-successive over- or under-relaxation (SOR) iteration. Numerical experiments, involving two linear and two nonlinear problems with their analytical solutions and two flow problems including the lid driven cavity and backward-facing step flows, are carried out to validate the performance of the newly proposed method. Numerical solutions of the driven cavity problem with different grid mesh sizes (maximum being 513513) for Reynolds numbers ranging from 0 to 17500 are obtained and compared with some of the accurate results available in the literature.