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: <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>
The systems of equations involved are tri-diagonal ([[symmetric matrix|symmetric]] [[Band matrix|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, 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.
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