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.

The method

Consider the linear diffusion equation in two dimensions,

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

The implicit Crank-Nicolson method produces the following finite difference equation:

${\displaystyle {u_{ij}^{n+1}-u_{ij}^{n} \over \Delta t}={1 \over 2}\left(\delta _{x}^{2}+\delta _{y}^{2}\right)\left(u_{ij}^{n+1}+u_{ij}^{n}\right)}$ 

where ${\displaystyle \delta _{p}^{2}}$  is the central difference operator for the p-coordinate After performing a stability analysis, it can be shown that this method will be stable as long as ${\displaystyle {\Delta t \over (\Delta x)^{2}}+{\Delta t \over (\Delta y)^{2}}<{1 \over 2}}$ .

This an unaffordable numerical stability criterion.

The idea behind the ADI method is to split the finite difference equations into two, one with the x-derivative taken implicitely and the next with the y-derivative taken implicitely, ${\displaystyle {u_{ij}^{n+1/2}-u_{ij}^{n} \over \Delta t/2}=\left(\delta _{x}^{2}u_{ij}^{n+1/2}+\delta _{y}^{2}u_{ij}^{n}\right)}$ 

${\displaystyle {u_{ij}^{n+1}-u_{ij}^{n+1/2} \over \Delta t/2}=\left(\delta _{x}^{2}u_{ij}^{n+1/2}+\delta _{y}^{2}u_{ij}^{n+1}\right)}$ 

It can be shown that this method is unconditionally stable. There are more refined ADI methods such as the methods of Douglas^[1], or the f-factor method^[2] which can be used for 3 or more dimensions.

1. Douglas, J. "Alternating direction methods for three space variables," Numerische Mathematik, Vol 4., pp 41-63 (1962)
2. Chang, M.J. et al. "Improved alternating-direction implicit method for solving transient three-dimensional heat diffusion problems", Numerical Heat Transfer, Vol 19, pp 69-84, (1991)