A note on the alternating direction implicit method for the numerical solution of heat flow problems

Abstract

1. Introduction. In companion papers [1; 6] recently Peaceman, Rachford, and the author introduced a finite difference technique called therein the alternating direction implicit method for approximating the solution of transient and permanent heat flow problems in two space variables. The validity of the method was established only in the case of a rectangular domain. Since then the procedure has been tested successfully on several more complex examples [4] without proof. The purpose of this short note is to prove in the case of non rectangular domains that the solution of the alternating direction method for the parabolic problem converges to the solution of the differential equation as the increments of the independent variables diminish in a proper manner, that the iterative adaptation for the elliptic problem converges to the solution of the Laplace difference equation, and to give an efficient choice of the parameter sequence involved in this iteration. 2. Parabolic problem. Let D be an open, connected set in the plane bounded by a curve C made up of closed polygons with sides parallel to the coordinate axes. Assume, moreover, that there exists a sequence