A New Iterative Procedure for the Solution of Sparse Systems of Linear Difference Equations

Summary

Partial differential equations of elliptic and parabolic type arise frequently in the mathematical analysis of many engineering problems. Frequently, these equations cannot be solved readily by analytical means. Consequently, efficient computer algorithms for the solution of sparse linear systems derived from the finite difference representations of a partial differential equation on a rectangular grid are of vital importance. In this paper, a new form of Successive Block Over-Relaxation in which the mesh points on a rectangular grid are ordered along successive peripherals of the domain is introduced. The coefficient matrix so obtained is sparse and can be shown to be block consistently ordered, possessing block Property A so that the whole S.O.R. theory is applicable. In order to solve each block of points, i.e., a peripheral circuit of the domain, a new solution algorithm for sparse sub-systems of equations is given. Improved rates of convergence to substantiate the theory and the new method, i.e., the Successive Peripheral Over-Relaxation method (S.P.O.R.) can be shown to be asymptotically equivalent to the S.2L.0.R. method. The new method is shown to cope well with special regions and the solution of the Torsion problem for a hollow square is given as illustration.

A new semi-direct method by this author was also presented at the conference. That method is based on a sparse matrix elimination algorithm, and is similar in concept to the Strongly Implicit Method of Stone (1968) Because of space limitations these results will be published elsewhere.