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Analysis of flow directed iterations. (English) Zbl 0743.76075

We consider some difference approximations to the convection diffusion equation and we treat block Gauss-Seidel iterations for the solution of these problems. We study the effect of the partitioning and ordering of the unknowns on the convergence of the Gauss-Seidel iterations. We find that, for convection dominated flow problems, the spectral radius of the iteration matrix is not an appropriate indicator of the convergence properties of the method; it is better to use a norm of the iteration matrix. Also, we find that sweeping the mesh in the direction of the underlying flow enhances the convergence of the Gauss-Seidel iterations. In one dimension it is not hard to devise an algorithm to implement this idea. In two dimensions, we give a general procedure to automate the partitioning and ordering phase of the solution process. The general procedure is described using the graph of the matrix.

MSC:

76R99 Diffusion and convection
65F10 Iterative numerical methods for linear systems
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