High-order Compact Difference Schemes And Multigrid Algorithms On Non-uniform Grids For The Two-Dimensional Incompressible Vorticity-stream Function Navier-Stokes Equations

The research on accurate, stable, and efficient numerical methods for incompressible Navier-Stokes (N-S) equations is an important content in the field of computational fluid dynamics. It is the key to achieve the accurate numerical simulation for various incompressible flow problems. In this paper, we are mainly aiming at studying the high order compact difference methods and the multigrid algorithms on non-uniform grids for solving the incompressible N-S equations.Firstly, we derive the high order compact difference scheme on non-uniform grids for the two-dimensional (2D) steady convection diffusion equation by the Taylor series expansion including compact approximations to the leading truncation error terms of the central difference scheme. Then a high order compact difference scheme for solving2D steady incompressible vorticity-stream function N-S equations is proposed. Based on the work for the steady problems, the second order backward Euler difference scheme is used to discretize the temporal derivative, a high order compact difference scheme for the2D unsteady convection diffusion equation and2D unsteady incompressible vorticity-stream function N-S equations is built. The present scheme is of the second order accuracy in time and the third to fourth order accuracy in space. In order to accelerate the convergence speed and improve the computational efficiency, multigrid method is employed to solve the linear algebraic system arising form the high order compact scheme. Finally, the Dirichlet boundary condition problems of2D convection diffusion equation and incompressible vorticity-stream function N-S equations with exact solution are conducted, and driven cavity flow problem is also simulated numerically. The computed results indicated that the present high order compact scheme on non-uniform grids can obtain more accurate solutions than those on uniform grids for the problems with boundary layers or great gradients. It’s also demonstrated that the multigrid method has higher convergence speed than the traditional iterative method. This fully verifies the accuracy, reliability and the high efficiency of the present method. The present method can be extended to solve other2D complicated incompressible flow problems.