Alternating-direction implicit method

In mathematics, the alternating direction implicit (ADI) method ^[1]is a finite difference method for solving parabolic and elliptic partial differential equations. It is most notably used to solve the problem of heat conduction or solving the diffusion equation in two or more dimensions.

The traditional method for solving the heat conduction equation is the Crank–Nicolson method. This method can be quite costly. The advantage of the ADI method is that the equations that have to be solved in every iteration have a simpler structure and are thus easier to solve.

The method

Consider the linear diffusion equation in two dimensions,

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

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 for any r.

A disadvantage of the Crank–Nicolson method is that the matrix in the above equation is banded with a band width that is generally quite large. This makes the solution of the equation quite costly.

The idea behind the ADI method is to split the finite difference equations into two, one with the x-derivative taken implicitly and the next with the y-derivative taken implicitly,

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

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

The systems of equations involved are symmetric and tridiagonal (banded with bandwidth 3), and thus cheap to solve by Choleski decomposition.

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

References

1. Peaceman, D. W. and Rachford, H. H., Jr., "The numerical solution of parabolic and elliptic differential equations", SIAM J. 3 (1955), 28-41, MR71874
2. Douglas, J. "Alternating direction methods for three space variables," Numerische Mathematik, Vol 4., pp 41-63 (1962)
3. 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)