Alternating-direction implicit method

Background

The alternating direction implicit (ADI) method is a finite difference method for solving differential equations. It is most notably used to solve the problem of heat conduction or solving the diffusion equation in 2 or more dimensions.

The traditional method for solving the heat conduction equation is the method of Crank-Nicolson. This method is implicit, but has an unaffordable stability criterion in 2 or more dimensions.

Example

Consider the linear diffusion equation in two dimensions,

${\displaystyle {\partial u \over \partial t}=k\left({\partial ^{2}u \over \partial x^{2}}+{\partial ^{2}u \over \partial y^{2}}+\right)=(u_{xx}+u_{yy}+u_{zz})\quad }$