The ADI Technique
(Alternating-Direction, Implicit)

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

tu = 2u - 4pGr .

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,

2u = 4pGr .

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.