Ordering Schemes for Parallel Processing of Certain Mesh Problems

In this work, some ordering schemes for mesh points are presented which enable algorithms such as the Gauss–Seidel or SOR iteration to be performed efficiently for the nine-point operator finite difference method on computers consisting of a two-dimensional grid of processors. Convergence results are presented for the discretization of $u_{xx} + u_{yy} $ on a uniform mesh over a square, showing that the spectral radius of the iteration for these orderings is no worse than that for the standard row by row ordering of mesh points. Further applications of these mesh point orderings to network problems, more general finite difference operators, and picture processing problems are noted.