Multigrid methods for higher-order finite difference schemes

Abstract

This thesis considers convergence analysis and effectiveness of multigrid method for certain higher order finite difference discretizations and 2nd order cell-centered finite difference of partial differential equations. First we deal with multigird algorithm for MVS(mean value scheme). MVS gets $h^6$ order accuracy through the symmetry of grid points for the Helmholtz equation. We apply multigrid algorithm to this discretization and prove convergence. We estimate the energy norm of prolongation operator and show that it strictly less than 1. According to the results in [15], we have V-cycle convergence. It is well known that the multigrid condition number for standard Laplace equation on the unit square is about 1.5. This is fast and most people believe that this is the fastest example. But, in numerical simulation, our multigrid scheme is faster than the standard one. This is in accordance with the result of our estimation. Next, we consider compact scheme. Compact scheme is a discretization on rectangular domain using both axis parallel finite differences and diagonal ones. So this scheme has 9-point stencil while the standard scheme is 5-point. This gives $h^6$(mesh size=h) order for smooth solutions and better accuracy than that $h^2$ of standard finite difference discretization. The multigrid convergence of the stand finite difference is considered in [15] and V-cycle convergence is proved. In this thesis, we consider the compact scheme and show the V-cycle multigrid convergence by energy norm estimation. In numerical simulation we can see that multigrid of compact scheme is faster than that of the standard one. Finally, we consider the cell-centered finite difference(CCFD) for elliptic partial differential equation with discontinuous coefficient. The cell-centered finite difference is a finite volume type of method and has been widely used by engineers due to its simplicity and local conservation property. As shown in [38], multigrid algorithm with certa...