Analysis of cyclic reduction for the numerical solution of three-dimensional convection-diffusion equations

• Main Author: Greif, Chen
• Format: Others
• Language: English
• Published: 2009
• Online Access: http://hdl.handle.net/2429/8505

This thesis deals with the numerical solution of convection-diffusion equations. In particular, the focus is on the analysis of applying one step of cyclic reduction to linear systems of equations which arise from finite difference discretization of steady-state three-dimensional convection-diffusion equations. The method is based on decoupling the unknowns and solving the resulting smaller linear systems using iterative methods. In three dimensions this procedure results in some loss of sparsity, compared to lower dimensions. Nevertheless, the resulting linear system has excellent numerical properties, is generally better conditioned than the original system, and gives rise to faster convergence of iterative solvers, and convergence in cases where solvers of the original system of equations fail to converge. The thesis starts with an overview of the equations that are solved and general properties of the resulting linear systems. Then, the unsymmetric discrete operator is derived and the structure of the cyclically reduced linear system is described. Several important aspects are analyzed in detail. The issue of orderings is addressed and a highly effective ordering strategy is presented. The complicated sparsity pattern of the matrix requires careful analysis; comprehensive convergence analysis for block stationary methods is provided, and the bounds on convergence rates are shown to be very tight. The computational work required to perform cyclic reduction and compute the solution of the linear system is discussed at length. Preconditioning techniques and various iterative solvers are considered. === Science, Faculty of === Mathematics, Department of === Graduate