| Summary: | 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
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