Solution of two-dimensional heat problems using the alternating direction method

Abstract

Finite difference approximations to the one-dimensional heat equation U[t] = U[xx] are used to introduce explicit and implicit difference equations. The convergence and stability of these equations is duscussed and these concepts are used to show the restrictions imposed on explicity equations. The Implicit Alternating Direction (IAD) method is introduced as a means for solution of the two-dimensional heat equation. Although the IAD method in its basic form is applicable only to parabolic problems, it is possible by slight modification to apply the method to elliptic problems. Two examples are used to illustrate the use of the IAD method for solution of parabolic and elliptic equations for a rectangular region. These examples include a work requirement comparison with other differences methods. A third example is given to show the applicability of the IAD method to non-rectangular regions. Results of these examples show that the IAD method is a conventient and accurate method when applied to both parabolic and elliptic partial differential equations and suggest applicability to a wide range of problems.