Ordering Effects on Relaxation Methods Applied to the Discrete Convection-Diffusion Equation
We present an analysis of relaxation methods for the discrete convection-diffusion equation based on norms of the iteration matrices. For one-dimensional problems, the results show how the performance of iterative solvers is affected by directions of flow associated with the underlying operator. In particular, for problems of size n, relaxation sweeps opposite the direction of flow incur 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. The one-dimensional analysis is also generalized to two-dimensional problems in the case where relaxation follows the flow.
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