A note on a generalisation of a method of Douglas

Abstract:

In this note, the high-order correct method of Douglas [1] for the diffusion equation in one space variable is extended to $q \leqq 3$ space variables. The resulting difference equations are then solved using the A. D. I. technique of Douglas and Gunn [3]. When $q = 2$, this method is equivalent to that of Mitchell and Fairweather [5] while $q = 3$ provides a method which is similar to Samarskiï’s method [6] and of higher accuracy than that of Douglas [2]. When the proposed methods are used to solve the diffusion equation with timeindependent boundary conditions, they have the advantage that no boundary modification (see [4]) is required to maintain accuracy.