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 Navier-Stokes equations in the stream function-vorticity 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 513×513) for Reynolds numbers ranging from 0 to 17500 are obtained and compared with some of the accurate results available in the literature.