Multigrid Method and High Order Compact Scheme for Solving Boundary Layer Problems on Nonuniform Grids

Jun Zhang

Laboratory for High Performance Scientific Computing and Computer Simulation
Department of Computer Science, University of Kentucky
Lexington, KY 40506-0046


A fourth order compact finite difference scheme (nine point compact difference scheme) and a multigrid cycling algorithm are employed to solve the two dimensional convection diffusion equations with boundary layers. The computational domain is first discretized on a nonuniform (stretched) grid to resolve the boundary layers. A grid transformation technique is used to map the nonuniform grid to a uniform one. The fourth order compact scheme is applied to the transformed uniform grid. A multigrid method is then used to solve the resulting linear system. We conduct experimental analyses to show how the grid stretching may affect the computed accuracy of the solutions from the fourth order compact scheme. We demonstrate that the grid stretching may affect the convergence rate of the multigrid method. Numerical experiments are used to show that a graded mesh and a grid transform are necessary to compute high accuracy solution with the fourth order compact scheme for convection diffusion problems with boundary layers. Accuracy comparisons between the standard (central and upwind) difference schemes and the present fourth order compact scheme are given. Special properties of the transformed convection diffusion equation that may affect multigrid convergence are investigated.