Consider a diffusion (heat transfer) equation of the form,

Given enough time, the system described by this equation will evolve to a steady-state "u" distribution such that ¶_tu = 0 and, hence, we arrive at a solution of the Poisson equation,

The idea behind the ADI scheme is to treat the Poisson equation as a diffusion equation, then adopt techniques that have been developed in connection with the solution of heat transfer problems to evolve the system to a steady-state. In this way an iterative technique is devised for the solution of the Poisson equation.

Reference:

• Peaceman, D. and Rachford, J. H. (1955) J. Soc. Indust. Appl. Math., 3, 28.
• Black, D. and Bodenheimer, P. (1975) The Astrophysical Journal, 199, 619.