Ordering Effects on Relaxation Methods Applied to the Discrete One-Dimensional Convection-Diffusion Equation

Abstract

The authors present an analysis of relaxation methods for the one-dimensional discrete convection-diffusion equation based on norms of the iteration matrices. In contrast to standard analytic techniques that use spectral radii, these results show how the performance of iterative solvers is affected by directions of flow associated with the underlying operator, and by orderings of the discrete grid points. In particular, for problems of size n, relaxation against the flow incurs a latency of approximately n steps in which convergence is slow, and red-black relaxation incurs a latency of approximately ${n / 2}$ steps. There is no latency associated with relaxation that follows the flow. These results are largely independent of the choice of discretization.