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▲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 ==
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The implicit Crank-Nicolson method produces the following finite difference equation:
: <math>{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)</math>
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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,
: <math>{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)</math>
: <math>{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)</math>
It can be shown that this method is unconditionally stable. There are more refined ADI methods such as the methods of Douglas<ref>Douglas, J. "Alternating direction methods for three space variables," Numerische Mathematik, Vol 4., pp 41-63 (1962)</ref>, or the f-factor method<ref>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)</ref> which can be used for
[[Category:Differential equations]]
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