On the Implementation of Parallel Implicit USSOR Methods for Solving Discrete Periodic Problems

David M. Young*, David R. Kincaid, and Wan Chen
Center for Numerical Analysis
The University of Texas at Austin

In this paper, we consider certain iterative methods
for solving discrete periodic problems based on the standard
5-point finite difference representation of the Poisson equation
in two dimensions. For a given positive integer m, our
procedure involves the implementation of m**2 single iterations
of the implicit USSOR (unsymmetric SOR) method based on the use
of m**2 pairs of relaxation factors, followed by the construction
of a linear combination of the results of those iterations.
It has previously been shown by the authors that the result
obtained by this process is the same as one would obtain by
using m iterations of the SSOR methods with varying relaxation
factor in sequence. (Thus, there is a potential saving in the
wall-clock time of a factor of m.)  To carry out the implicit USSOR method, 
we consider various procedures including the (non-implicit) SOR method. 
The choice of the optimum relaxation factor for the SOR method and 
the estimation of the corresponding convergence properties is greatly
simplified since a related matrix is p-cyclic.