Line Iterative Methods for Cyclically Reduced Discrete Convection-Diffusion Problems

An analytic and empirical study of line iterative methods for solving the discrete convection-diffusion equation is performed. The methodology consists of performing one step of the cyclic reduction method, followed by iteration on the resulting reduced system using line orderings of the reduced grid. Two classes of iterative methods are considered: block stationary methods, such as the block Gauss-Seidel and SOR methods, and preconditioned generalized minimum residual methods with incomplete LU preconditioners. New analysis extends convergence bounds for constant coefficient problems to problems with separable variable coefficients. In addition, analytic results show that iterative methods based on incomplete LU preconditioners have faster convergence rates than block Jacobi relaxation methods. Numerical experiments examine additional properties of the two classes of methods, including the effects of direction of flow, discretization, and grid ordering on performance.