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A disadvantage of the Crank–Nicolson method is that the matrix in the above equation is [[band matrix|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
: <math>{u_{ij}^{n+1/2}-u_{ij}^n\over \Delta t/2} =
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The systems of equations involved are [[symmetric matrix|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<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,Jefferies, 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 three or more dimensions.
== References ==
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