Alternating direction implicit method

In mathematics, the alternating direction implicit (ADI) method [Peaceman, D. W. and Rachford, H. H., Jr., "The numerical solution of parabolic and elliptic differential equations",SIAM J. 3 (1955), 28-41, [http://www.ams.org/mathscinet-getitem?mr=71874 MR71874] ] 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 method of Crank-Nicolson. This method is implicit, but has an unaffordable stability criterion in two or more dimensions.

The method

Consider the linear diffusion equation in two dimensions,

: $\{partial\; uover\; partial\; t\}\; =\; left(\{partial^2\; uover\; partial\; x^2\; \}\; +\{partial^2\; uover\; partial\; y^2\; \}\; ight)\; =\; (\; u\_\{xx\}\; +\; u\_\{yy\}\; )\; quad$

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

: $\{u\_\{ij\}^\{n+1\}-u\_\{ij\}^nover\; Delta\; t\}\; =\; \{1\; over\; 2\}left(delta\_x^2+delta\_y^2\; ight)left(u\_\{ij\}^\{n+1\}+u\_\{ij\}^n\; ight)$

where $delta\_p^2$ 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

:

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 implicitly and the next with the y-derivative taken implicitly,

: $\{u\_\{ij\}^\{n+1/2\}-u\_\{ij\}^nover\; Delta\; t/2\}\; =\; left(delta\_x^2\; u\_\{ij\}^\{n+1/2\}+delta\_y^2\; u\_\{ij\}^\{n\}\; ight)$

: $\{u\_\{ij\}^\{n+1\}-u\_\{ij\}^\{n+1/2\}over\; Delta\; t/2\}\; =\; left(delta\_x^2\; u\_\{ij\}^\{n+1/2\}+delta\_y^2\; u\_\{ij\}^\{n+1\}\; ight)$

The systems of equations involved are tri-diagonal (symmetric 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 [Douglas, J. "Alternating direction methods for three space variables," Numerische Mathematik, Vol 4., pp 41-63 (1962)] , or the f-factor method [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)] which can be used for three or more dimensions.

References