A STRONGLY IMPLICIT SOLVER FOR TWO-DIMENSIONAL ELLIPTIC DIFFERENTIAL EQUATIONS

Abstract

A strongly implicit solver is proposed in the present investigation for solving a large set of algebraic equations that arise from a discretization of an elliptic differential equation. Based on the guessed θ_NW and θ_SE values (solutions at the northwest and the southeast corners of a computational cell), the nine-diagonal coefficient matrix is factored in terms of a lower and an upper triangular matrix with only seven nonzero diagonals. The solution procedure then is iterated with a successive overrelaxation (SOR) factor until the solution converges within a prescribed tolerance. In the present solver, there is no need to evaluate the residual for the guessed solution. The CPU time thus is reduced a great amount for a single iteration. In addition, the storage is only one-half as large as that required by the SIP solver because the original matrix is no longer needed after it is factored. The convergence rate of the SIP solver is very sensitive to the cancellation parameter a. The present solver needs no partial cancellation procedure, so it does not pose this difficulty. Through examples, the present solver is seen to be particularly good for use in solving convective heat transfer problems.

Notes

The author thanks the National Science Council of the Republic of China in Taiwan for the financial support of this work through project NSC76-0401-E007-11.