A fourth-order compact finite difference scheme for the steady stream function-vorticity formulation of the NavierStokes/Boussinesq equations

A fourth-order compact finite difference scheme on the nine-point 2D stencil is formulated for solving the steady-state NavierStokes/Boussinesq equations for two-dimensional, incompressible fluid flow and heat transfer using the stream function-vorticity formulation. The main feature of the new fourth-order compact scheme is that it allows point-successive overrelaxation (SOR) or point-successive underrelaxation iteration for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. Numerical solutions are obtained for the model problem of natural convection in a square cavity with benchmark solutions and compared with some of the accurate results available in the literature.