High-order Compact Difference Schemes And Multigrid Algorithms On Nonuniform Grids For The Two Dimensional Unsteady Vorticity-velocity Navier-Stokes Equations

Study of the numerical solution method for the incompressible Navier-Stokes equations (N-S) is a focus in the area of computational fluid dynamics. And finite difference method is one of the most widely applicated methods for solving these equations. So far, many accurate, stable and efficient methods have been developed. However, many of them are construsted on uniform grids, but few of them are built on nonuniform grids. On the other hand, the multigrid method is a computational method to realize a fast solution for the algebraic equations while many kinds of partial differential equations are discretized. In this paper, we are aiming at the research on a high order compact difference scheme for unsteady vorticity-velocity N-S equations on nonuniform grids and the multigrid algorithm. A high order compact difference scheme of unsteady vorticity-velocity N-S equations is constructed which is based on nonuniform grids by using the high order compact (HOC) difference schemes of unsteady convection diffusion equation that has existed, and then multigrid algorithm is combined to solve the algebraic equations deriving from it. In order to confirm the effectiveness of the present algorithm, numerical experiments for the problems that contain great gradient or boundary layers and the driven cavity flow problem are carried out. The computed results show that the numerical results of present method are in good agreement with the exact or benchmark solutions, so it abundantly shows accuracy of the present method. Meanwhile, the accelerating effect of the multigrid method are confirmed that is very significant.