Here let us now consider a plate whose initial conditions are specified .
FTCS scheme
So Let us discretize the governing 2 -Dimensional Parabolic PDE using FTCS scheme .
$$\frac{T_{i,j}^{n+1}-T_{i,j}^{n}}{\Delta t} = \alpha \left (\frac{T_{i+1,j}^{n}-2T_{i,j}^{n}+T_{i-1,j}^{n}}{\Delta x ^2} +\frac{T_{i,j+1}^{n}-2T_{i,j}^{n}+T_{i,j-1}^{n}}{\Delta y ^2} \right ) $$
Since this is an explicit scheme it becomes easy to calculate the solutions of $T$ as a function of time.(ie)
$$T_{i,j}^{n+1} = T_{i,j}^{n}+ \Delta t \alpha \left (\frac{T_{i+1,j}^{n}-2T_{i,j}^{n}+T_{i-1,j}^{n}}{\Delta x ^2} +\frac{T_{i,j+1}^{n}-2T_{i,j}^{n}+T_{i,j-1}^{n}}{\Delta y ^2} \right )$$
For computational ease $\Delta x = \Delta y$ can be used. Now let us see the stability criterion of the scheme below.
ADI scheme
The ADI (alternate direction implicit method) is the execution of the implicit schemes at one direction at time steps $n,n +1/2$
$$\frac{T_{i,j}^{n+1/2}-T_{i,j}^{n}}{\Delta t} = \alpha \left (\frac{T_{i+1,j}^{n+1/2}-2T_{i,j}^{n+1/2}+T_{i-1,j}^{n+1/2}}{\Delta x ^2} +\frac{T_{i,j+1}^{n}-2T_{i,j}^{n}+T_{i,j-1}^{n}}{\Delta y ^2} \right ) $$
$$\frac{T_{i,j}^{n+1}-T_{i,j}^{n+1/2}}{\Delta t} = \alpha \left (\frac{T_{i+1,j}^{n+1}-2T_{i,j}^{n+1}+T_{i-1,j}^{n+1}}{\Delta x ^2} +\frac{T_{i,j+1}^{n+1/2}-2T_{i,j}^{n+1/2}+T_{i,j-1}^{n+1/2}}{\Delta y ^2} \right ) $$
The above equations as can be seen that it forms a tridiagonal matrix which can easily be solved by TDMA algorithm . The first equation is solved for finding the solutions for intermediate time step $n +1/2$ then this is used to find the solution at time step $n+1$ .
FTCS scheme
So Let us discretize the governing 2 -Dimensional Parabolic PDE using FTCS scheme .
$$\frac{T_{i,j}^{n+1}-T_{i,j}^{n}}{\Delta t} = \alpha \left (\frac{T_{i+1,j}^{n}-2T_{i,j}^{n}+T_{i-1,j}^{n}}{\Delta x ^2} +\frac{T_{i,j+1}^{n}-2T_{i,j}^{n}+T_{i,j-1}^{n}}{\Delta y ^2} \right ) $$
Since this is an explicit scheme it becomes easy to calculate the solutions of $T$ as a function of time.(ie)
$$T_{i,j}^{n+1} = T_{i,j}^{n}+ \Delta t \alpha \left (\frac{T_{i+1,j}^{n}-2T_{i,j}^{n}+T_{i-1,j}^{n}}{\Delta x ^2} +\frac{T_{i,j+1}^{n}-2T_{i,j}^{n}+T_{i,j-1}^{n}}{\Delta y ^2} \right )$$
For computational ease $\Delta x = \Delta y$ can be used. Now let us see the stability criterion of the scheme below.
Stability Condition
It can be derived to show that the stability for the 2 D heat conduction equation is similar to that of the 1-D heat equation ,hence
$$ \alpha \Delta (t) \left (\frac{1}{(\Delta x )^2} + \frac{1}{(\Delta x )^2} \right )\leq \frac{1}{2}$$
Check Computational Fluid Dynamics by Klaus A. Hoffmann (Author), Steve T. Chiang (Author) for reference.
This can be further reduced to
$$ \alpha \Delta (t) \left (\frac{1}{(\Delta x )^2} \right )\leq \frac{1}{4}$$ when $\Delta x = \Delta y$
BTCS scheme
The same can be done in BTCS scheme to obtain
$$\frac{T_{i,j}^{n+1}-T_{i,j}^{n}}{\Delta t} = \alpha \left (\frac{T_{i+1,j}^{n+1}-2T_{i,j}^{n+1}+T_{i-1,j}^{n+1}}{\Delta x ^2} +\frac{T_{i,j+1}^{n+1}-2T_{i,j}^{n+1}+T_{i,j-1}^{n+1}}{\Delta y ^2} \right ) $$
The above equation as can be seen that it forms a pentadiagonal matrix which can be solved by matrix inversion which is computationally expensive ,hence this scheme wont be used most of the time. Hence emerges the ADI scheme
The ADI (alternate direction implicit method) is the execution of the implicit schemes at one direction at time steps $n,n +1/2$
$$\frac{T_{i,j}^{n+1/2}-T_{i,j}^{n}}{\Delta t} = \alpha \left (\frac{T_{i+1,j}^{n+1/2}-2T_{i,j}^{n+1/2}+T_{i-1,j}^{n+1/2}}{\Delta x ^2} +\frac{T_{i,j+1}^{n}-2T_{i,j}^{n}+T_{i,j-1}^{n}}{\Delta y ^2} \right ) $$
$$\frac{T_{i,j}^{n+1}-T_{i,j}^{n+1/2}}{\Delta t} = \alpha \left (\frac{T_{i+1,j}^{n+1}-2T_{i,j}^{n+1}+T_{i-1,j}^{n+1}}{\Delta x ^2} +\frac{T_{i,j+1}^{n+1/2}-2T_{i,j}^{n+1/2}+T_{i,j-1}^{n+1/2}}{\Delta y ^2} \right ) $$
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