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SUMMARYA fourth-order compact ÿnite di erence scheme on the nine-point 2D stencil is formulated for solving the steady-state Navier-Stokes=Boussinesq equations for two-dimensional, incompressible uid ow 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.

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SUMMARYOn the basis of the projection method, a higher order compact finite difference algorithm, which possesses a good spatial behavior, is developed for solving the 2D unsteady incompressible Navier-Stokes equations in primitive variable. The present method is established on a staggered grid system and is at least third-order accurate in space. A third-order accurate upwind compact difference approximation is used to discretize the non-linear convective terms, a fourth-order symmetrical compact difference approximation is used to discretize the viscous terms, and a fourth-order compact difference approximation on a cell-centered mesh is used to discretize the first derivatives in the continuity equation. The pressure Poisson equation is approximated using a fourth-order compact difference scheme constructed currently on the nine-point 2D stencil. New fourth-order compact difference schemes for explicit computing of the pressure gradient are also developed on the nine-point 2D stencil. For the assessment of the effectiveness and accuracy of the method, particularly its spatial behavior, a problem with analytical solution and another one with a steep gradient are numerically solved. Finally, steady and unsteady solutions for the lid-driven cavity flow are also used to assess the efficiency of this algorithm.

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