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where <math>\delta_p^2</math> 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
<math>{\Delta t\over (\Delta x)^2+\Delta t\over (\Delta y)^2} < {1 \over 2}</math>.
This is a high stability criterion.
The idea behind the ADI method is to split the finite difference equations into two, one with the x-derivative taken implicitely and the next with the y-derivative taken implicitely,
<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 3 or more dimensions.
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