Team:Valencia Biocampus/Demonstration/Diffusion2

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Description of the numerical method

The Crank-Nicolson method is obtained by using a forward difference in time ($t_k$) and central differences in space $(x_i, y_j)$. However, by giving discrete values for time and space $(t_k, x_i, y_j)$ to our Crank-Nicolson method and zeroing boundary conditions (since we have to define some limit for the plate) and also setting initial conditions to a specific initial distribution, we achieved the matrix recursive formula: $$[\mathcal{C}^{k+1}] = \mathcal{M}\;[\mathcal{C}^k]$$ Where $\mathcal{M}$ is a block tridiagonal matrix depending on the parameters and the velocities in $x$ and $y$ directions at some points in space.
1) The first step is to discretize our differential equation, replacing the time derivatives by the approximate forward differences and the space derivatives by central differences, being $\;q\;$ and $\;h\;$ the time and space step, respectively:

$$\; 2 \; \frac{C^{k+1}_{i,j} - C^{k}_{i,j}}{q}\;= \\ \;D \; \left(\frac{C^{k+1}_{i+1,j} - 2C^{k+1}_{i,j}+C^{k+1}_{i-1,j}}{h^2} + \frac{C^{k+1}_{i,j+1} - 2C^{k+1}_{i,j}+C^{k+1}_{i,j-1}}{h^2}\right) - \left(\frac{\nu_{x_{i+1,j}} - \nu_{x_{i-1,j}}}{2h} \; C^{k+1}_{i,j} + \nu_{x_{i,j}} \; \frac{C^{k+1}_{i+1,j} - C^{k+1}_{i-1,j}}{2h} + \frac{\nu_{y_{i,j+1}} - \nu_{y_{i,j-1}}}{2h} \; C^{k+1}_{i,j} + \nu_{y_{i,j}} \; \frac{C^{k+1}_{i,j+1} - C^{k+1}_{i,j-1}}{2h}\right) \\ + D \; \left(\frac{C^{k}_{i+1,j} - 2C^{k}_{i,j}+C^{k}_{i-1,j}}{h^2} + \frac{C^{k}_{i,j+1} - 2C^{k}_{i,j}+C^{k}_{i,j-1}}{h^2}\right) - \left(\frac{\nu_{x_{i+1,j}} - \nu_{x_{i-1,j}}}{2h} \; C^{k}_{i,j} + \nu_{x_{i,j}} \; \frac{C^{k}_{i+1,j} - C^{k}_{i-1,j}}{2h} + \frac{\nu_{y_{i,j+1}} - \nu_{y_{i,j-1}}}{2h} \; C^{k}_{i,j} + \nu_{y_{i,j}} \; \frac{C^{k}_{i,j+1} - C^{k}_{i,j-1}}{2h}\right)$$
Then, placing all (k+1) terms at left and (k) terms at right and recalling the constants $\;\lambda \; = \; \frac{q \; D}{2 \; h^2}\;$ and $\;\mu \; = \; \frac{q}{4 \; h}\;$ it results a more handy equation:

$$ C^{k+1}_{i+1,j} \; \left(- \lambda + \nu_{x_{i,j}} \; \mu \right) + C^{k+1}_{i,j+1} \; \left(- \lambda + \nu_{y_{i,j}} \; \mu \right) + C^{k+1}_{i,j} \; \left(1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{i,j}} + \nu^{(0,1)}_{{y}_{i,j}}\right) + C^{k+1}_{i-1,j} \; \left(- \lambda - \nu_{x_{i,j}} \; \mu \right) + C^{k+1}_{i,j-1} \; \left(- \lambda - \nu_{y_{i,j}} \; \mu \right) \; = \\ \; C^{k}_{i+1,j} \; \left(\lambda - \nu_{x_{i,j}} \; \mu \right) + C^{k}_{i,j+1} \; \left(\lambda - \nu_{y_{i,j}} \; \mu \right) + C^{k}_{i,j} \; \left(1 - 4 \; \lambda - \nu^{(1,0)}_{{x}_{i,j}} - \nu^{(0,1)}_{{y}_{i,j}}\right) + C^{k}_{i-1,j} \; \left(\lambda + \nu_{x_{i,j}} \; \mu \right) + C^{k}_{i,j-1} \; \left(\lambda + \nu_{y_{i,j}} \; \mu \right) $$
2) For being able to compute the method in a matrix way, we were forced to do a small cencrete example, in order to extrapolate then the procedure into a general method. For that, we meshed the space in three steps each coordinate, so that, i=1,2,3 and j=1,2,3:

$ i=1,\;j=1 $ $$ C^{k+1}_{2,1} \; \left(- \lambda + \nu_{x_{1,1}} \; \mu \right) + C^{k+1}_{1,2} \; \left(- \lambda + \nu_{y_{1,1}} \; \mu \right) + C^{k+1}_{1,1} \; \left(1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{1,1}} + \nu^{(0,1)}_{{y}_{1,1}}\right) + C^{k+1}_{0,1} \; \left(- \lambda - \nu_{x_{1,1}} \; \mu \right) + C^{k+1}_{1,0} \; \left(- \lambda - \nu_{y_{1,1}} \; \mu \right) \; = \\ \; C^{k}_{2,1} \; \left(\lambda - \nu_{x_{1,1}} \; \mu \right) + C^{k}_{1,2} \; \left(\lambda - \nu_{y_{1,1}} \; \mu \right) + C^{k}_{1,1} \; \left(1 - 4 \; \lambda - \nu^{(1,0)}_{{x}_{1,1}} - \nu^{(0,1)}_{{y}_{1,1}}\right) + C^{k}_{0,1} \; \left(\lambda + \nu_{x_{1,1}} \; \mu \right) + C^{k}_{1,0} \; \left(\lambda + \nu_{y_{1,1}} \; \mu \right) $$ $ i=1,\;j=2 $ $$ C^{k+1}_{2,2} \; \left(- \lambda + \nu_{x_{1,2}} \; \mu \right) + C^{k+1}_{1,3} \; \left(- \lambda + \nu_{y_{1,2}} \; \mu \right) + C^{k+1}_{1,2} \; \left(1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{1,2}} + \nu^{(0,1)}_{{y}_{1,2}}\right) + C^{k+1}_{0,2} \; \left(- \lambda - \nu_{x_{1,2}} \; \mu \right) + C^{k+1}_{1,1} \; \left(- \lambda - \nu_{y_{1,2}} \; \mu \right) \; = \\ \; C^{k}_{2,2} \; \left(\lambda - \nu_{x_{1,2}} \; \mu \right) + C^{k}_{1,3} \; \left(\lambda - \nu_{y_{1,2}} \; \mu \right) + C^{k}_{1,2} \; \left(1 - 4 \; \lambda - \nu^{(1,0)}_{{x}_{1,2}} - \nu^{(0,1)}_{{y}_{1,2}}\right) + C^{k}_{0,2} \; \left(\lambda + \nu_{x_{1,2}} \; \mu \right) + C^{k}_{1,1} \; \left(\lambda + \nu_{y_{1,2}} \; \mu \right) $$ $ i=1,\;j=3 $ $$ C^{k+1}_{2,3} \; \left(- \lambda + \nu_{x_{1,3}} \; \mu \right) + C^{k+1}_{1,4} \; \left(- \lambda + \nu_{y_{1,3}} \; \mu \right) + C^{k+1}_{1,3} \; \left(1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{1,3}} + \nu^{(0,1)}_{{y}_{1,3}}\right) + C^{k+1}_{0,3} \; \left(- \lambda - \nu_{x_{1,3}} \; \mu \right) + C^{k+1}_{1,2} \; \left(- \lambda - \nu_{y_{1,3}} \; \mu \right) \; = \\ \; C^{k}_{2,3} \; \left(\lambda - \nu_{x_{1,3}} \; \mu \right) + C^{k}_{1,4} \; \left(\lambda - \nu_{y_{1,3}} \; \mu \right) + C^{k}_{1,3} \; \left(1 - 4 \; \lambda - \nu^{(1,0)}_{{x}_{1,3}} - \nu^{(0,1)}_{{y}_{1,3}}\right) + C^{k}_{0,3} \; \left(\lambda + \nu_{x_{1,3}} \; \mu \right) + C^{k}_{1,2} \; \left(\lambda + \nu_{y_{1,3}} \; \mu \right) $$ $ i=2,\;j=1 $ $$ C^{k+1}_{3,1} \; \left(- \lambda + \nu_{x_{2,1}} \; \mu \right) + C^{k+1}_{2,2} \; \left(- \lambda + \nu_{y_{2,1}} \; \mu \right) + C^{k+1}_{2,1} \; \left(1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{2,1}} + \nu^{(0,1)}_{{y}_{2,1}}\right) + C^{k+1}_{1,1} \; \left(- \lambda - \nu_{x_{2,1}} \; \mu \right) + C^{k+1}_{2,0} \; \left(- \lambda - \nu_{y_{2,1}} \; \mu \right) \; = \\ \; C^{k}_{3,1} \; \left(\lambda - \nu_{x_{2,1}} \; \mu \right) + C^{k}_{2,2} \; \left(\lambda - \nu_{y_{2,1}} \; \mu \right) + C^{k}_{2,1} \; \left(1 - 4 \; \lambda - \nu^{(1,0)}_{{x}_{2,1}} - \nu^{(0,1)}_{{y}_{2,1}}\right) + C^{k}_{1,1} \; \left(\lambda + \nu_{x_{2,1}} \; \mu \right) + C^{k}_{2,0} \; \left(\lambda + \nu_{y_{2,1}} \; \mu \right) $$ $ i=2,\;j=2 $ $$ C^{k+1}_{3,2} \; \left(- \lambda + \nu_{x_{2,2}} \; \mu \right) + C^{k+1}_{2,3} \; \left(- \lambda + \nu_{y_{2,2}} \; \mu \right) + C^{k+1}_{2,2} \; \left(1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{2,2}} + \nu^{(0,1)}_{{y}_{2,2}}\right) + C^{k+1}_{1,2} \; \left(- \lambda - \nu_{x_{2,2}} \; \mu \right) + C^{k+1}_{2,1} \; \left(- \lambda - \nu_{y_{2,2}} \; \mu \right) \; = \\ \; C^{k}_{3,2} \; \left(\lambda - \nu_{x_{2,2}} \; \mu \right) + C^{k}_{2,3} \; \left(\lambda - \nu_{y_{2,2}} \; \mu \right) + C^{k}_{2,2} \; \left(1 - 4 \; \lambda - \nu^{(1,0)}_{{x}_{2,2}} - \nu^{(0,1)}_{{y}_{2,2}}\right) + C^{k}_{1,2} \; \left(\lambda + \nu_{x_{2,2}} \; \mu \right) + C^{k}_{2,1} \; \left(\lambda + \nu_{y_{2,2}} \; \mu \right) $$ $ i=2,\;j=3 $ $$ C^{k+1}_{3,3} \; \left(- \lambda + \nu_{x_{2,3}} \; \mu \right) + C^{k+1}_{2,4} \; \left(- \lambda + \nu_{y_{2,3}} \; \mu \right) + C^{k+1}_{2,3} \; \left(1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{2,3}} + \nu^{(0,1)}_{{y}_{2,3}}\right) + C^{k+1}_{1,3} \; \left(- \lambda - \nu_{x_{2,3}} \; \mu \right) + C^{k+1}_{2,2} \; \left(- \lambda - \nu_{y_{2,3}} \; \mu \right) \; = \\ \; C^{k}_{3,3} \; \left(\lambda - \nu_{x_{2,3}} \; \mu \right) + C^{k}_{2,4} \; \left(\lambda - \nu_{y_{2,3}} \; \mu \right) + C^{k}_{2,3} \; \left(1 - 4 \; \lambda - \nu^{(1,0)}_{{x}_{2,3}} - \nu^{(0,1)}_{{y}_{2,3}}\right) + C^{k}_{1,3} \; \left(\lambda + \nu_{x_{2,3}} \; \mu \right) + C^{k}_{2,2} \; \left(\lambda + \nu_{y_{2,3}} \; \mu \right) $$ $ i=3,\;j=1 $ $$ C^{k+1}_{4,1} \; \left(- \lambda + \nu_{x_{3,1}} \; \mu \right) + C^{k+1}_{3,2} \; \left(- \lambda + \nu_{y_{3,1}} \; \mu \right) + C^{k+1}_{3,1} \; \left(1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{3,1}} + \nu^{(0,1)}_{{y}_{3,1}}\right) + C^{k+1}_{2,1} \; \left(- \lambda - \nu_{x_{3,1}} \; \mu \right) + C^{k+1}_{3,0} \; \left(- \lambda - \nu_{y_{3,1}} \; \mu \right) \; = \\ \; C^{k}_{4,1} \; \left(\lambda - \nu_{x_{3,1}} \; \mu \right) + C^{k}_{3,2} \; \left(\lambda - \nu_{y_{3,1}} \; \mu \right) + C^{k}_{3,1} \; \left(1 - 4 \; \lambda - \nu^{(1,0)}_{{x}_{3,1}} - \nu^{(0,1)}_{{y}_{3,1}}\right) + C^{k}_{2,1} \; \left(\lambda + \nu_{x_{3,1}} \; \mu \right) + C^{k}_{3,0} \; \left(\lambda + \nu_{y_{3,1}} \; \mu \right) $$ $ i=3,\;j=2 $ $$ C^{k+1}_{4,2} \; \left(- \lambda + \nu_{x_{3,2}} \; \mu \right) + C^{k+1}_{3,3} \; \left(- \lambda + \nu_{y_{3,2}} \; \mu \right) + C^{k+1}_{3,2} \; \left(1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{3,2}} + \nu^{(0,1)}_{{y}_{3,2}}\right) + C^{k+1}_{2,2} \; \left(- \lambda - \nu_{x_{3,2}} \; \mu \right) + C^{k+1}_{3,1} \; \left(- \lambda - \nu_{y_{3,2}} \; \mu \right) \; = \\ \; C^{k}_{4,2} \; \left(\lambda - \nu_{x_{3,2}} \; \mu \right) + C^{k}_{3,3} \; \left(\lambda - \nu_{y_{3,2}} \; \mu \right) + C^{k}_{3,2} \; \left(1 - 4 \; \lambda - \nu^{(1,0)}_{{x}_{3,2}} - \nu^{(0,1)}_{{y}_{3,2}}\right) + C^{k}_{2,2} \; \left(\lambda + \nu_{x_{3,2}} \; \mu \right) + C^{k}_{3,1} \; \left(\lambda + \nu_{y_{3,2}} \; \mu \right) $$ $ i=3,\;j=3 $ $$ C^{k+1}_{4,3} \; \left(- \lambda + \nu_{x_{3,3}} \; \mu \right) + C^{k+1}_{3,4} \; \left(- \lambda + \nu_{y_{3,3}} \; \mu \right) + C^{k+1}_{3,3} \; \left(1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{3,3}} + \nu^{(0,1)}_{{y}_{3,3}}\right) + C^{k+1}_{2,3} \; \left(- \lambda - \nu_{x_{3,3}} \; \mu \right) + C^{k+1}_{3,2} \; \left(- \lambda - \nu_{y_{3,3}} \; \mu \right) \; = \\ \; C^{k}_{4,3} \; \left(\lambda - \nu_{x_{3,3}} \; \mu \right) + C^{k}_{3,4} \; \left(\lambda - \nu_{y_{3,3}} \; \mu \right) + C^{k}_{3,3} \; \left(1 - 4 \; \lambda - \nu^{(1,0)}_{{x}_{3,3}} - \nu^{(0,1)}_{{y}_{3,3}}\right) + C^{k}_{2,3} \; \left(\lambda + \nu_{x_{3,3}} \; \mu \right) + C^{k}_{3,2} \; \left(\lambda + \nu_{y_{3,3}} \; \mu \right) $$
Our goal was to rewrite the ecuations system, in the form:

$$ \underline{\underline{A}} \; \underline{C^{k+1}} \; = \; \underline{\underline{B}} \; \underline{C^{k}} $$
So, in that moment, we could finally put all that in a matrix form, being all $\;C^{k+1}_{i,j}\;$ the unknowns, and $\;C^{k}_{i,j}\;$ and rest of the constants, the knowns, and making zero all terms that contained i=0 or i=4 or j=0 or j=4, that are our boundary conditions. Renamed $\;\alpha \; = \; 1 + 4 \; \lambda\;$, $\;\beta \; = \; 1 - 4 \; \lambda\;$ and $\;V_{i,j}\;=\;\nu^{(1,0)}_{{x}_{i,j}} + \nu^{(0,1)}_{{y}_{i,j}}$:

$$ \begin{bmatrix} \begin{array}{ccc|ccc|ccc} \alpha + \nu^{(1,0)}_{{x}_{1,1}} + \nu^{(0,1)}_{{y}_{1,1}} & - \lambda + \nu_{y_{1,1}} \; \mu & 0 & - \lambda + \nu_{x_{1,1}} \; \mu & 0 & 0 & 0 & 0 & 0\\ - \lambda - \nu_{y_{1,2}} \; \mu & \alpha + \nu^{(1,0)}_{{x}_{1,2}} + \nu^{(0,1)}_{{y}_{1,2}} & - \lambda + \nu_{y_{1,2}} \; \mu & 0 & - \lambda + \nu_{x_{1,2}} \; \mu & 0 & 0 & 0 & 0\\ 0 & - \lambda - \nu_{y_{1,3}} \; \mu & \alpha + \nu^{(1,0)}_{{x}_{1,3}} + \nu^{(0,1)}_{{y}_{1,3}} & 0 & 0 & - \lambda + \nu_{x_{1,3}} \; \mu & 0 & 0 & 0\\ \hline\\ - \lambda - \nu_{x_{2,1}} \; \mu & 0 & 0 & \alpha + \nu^{(1,0)}_{{x}_{2,1}} + \nu^{(0,1)}_{{y}_{2,1}} & - \lambda + \nu_{y_{2,1}} \; \mu & 0 & - \lambda + \nu_{x_{2,1}} \; \mu & 0 & 0\\ 0 & - \lambda - \nu_{x_{2,2}} \; \mu & 0 & - \lambda - \nu_{y_{2,2}} \; \mu & \alpha + \nu^{(1,0)}_{{x}_{2,2}} + \nu^{(0,1)}_{{y}_{2,2}} & - \lambda + \nu_{y_{2,2}} \; \mu & 0 & - \lambda + \nu_{x_{2,2}} \; \mu & 0\\ 0 & 0 & - \lambda - \nu_{x_{2,3}} \; \mu & 0 & - \lambda - \nu_{y_{2,3}} \; \mu & \alpha + \nu^{(1,0)}_{{x}_{2,3}} + \nu^{(0,1)}_{{y}_{2,3}} & 0 & 0 & - \lambda + \nu_{x_{2,3}} \; \mu\\ \hline\\ 0 & 0 & 0 & - \lambda - \nu_{x_{3,1}} \; \mu & 0 & 0 & \alpha + \nu^{(1,0)}_{{x}_{3,1}} + \nu^{(0,1)}_{{y}_{3,1}} & - \lambda + \nu_{y_{3,1}} \; \mu & 0\\ 0 & 0 & 0 & 0 & - \lambda - \nu_{x_{3,2}} \; \mu & 0 & - \lambda - \nu_{y_{3,2}} \; \mu & \alpha + \nu^{(1,0)}_{{x}_{3,2}} + \nu^{(0,1)}_{{y}_{3,2}} & - \lambda + \nu_{y_{3,2}} \; \mu\\ 0 & 0 & 0 & 0 & 0 & - \lambda - \nu_{x_{3,3}} \; \mu & 0 & - \lambda - \nu_{y_{3,3}} \; \mu & \alpha + \nu^{(1,0)}_{{x}_{3,3}} + \nu^{(0,1)}_{{y}_{3,3}} \end{array} \end{bmatrix}\begin{bmatrix} C^{k+1}_{1,1}\\C^{k+1}_{1,2}\\C^{k+1}_{1,3}\\\hline\\C^{k+1}_{2,1}\\C^{k+1}_{2,2}\\C^{k+1}_{2,3}\\\hline\\C^{k+1}_{3,1}\\C^{k+1}_{3,2}\\C^{k+1}_{3,3}\end{bmatrix}=$$
$$\begin{bmatrix} \begin{array}{ccc|ccc|ccc} \beta - \nu^{(1,0)}_{{x}_{1,1}} - \nu^{(0,1)}_{{y}_{1,1}} & \lambda - \nu_{y_{1,1}} \; \mu & 0 & \lambda - \nu_{x_{1,1}} \; \mu & 0 & 0 & 0 & 0 & 0\\ \lambda + \nu_{y_{1,2}} \; \mu & \beta - \nu^{(1,0)}_{{x}_{1,2}} - \nu^{(0,1)}_{{y}_{1,2}} & \lambda - \nu_{y_{1,2}} \; \mu & 0 & \lambda - \nu_{x_{1,2}} \; \mu & 0 & 0 & 0 & 0\\ 0 & \lambda + \nu_{y_{1,3}} \; \mu & \beta - \nu^{(1,0)}_{{x}_{1,3}} - \nu^{(0,1)}_{{y}_{1,3}} & 0 & 0 & \lambda - \nu_{x_{1,3}} \; \mu & 0 & 0 & 0\\ \hline\\ \lambda + \nu_{x_{2,1}} \; \mu & 0 & 0 & \beta - \nu^{(1,0)}_{{x}_{2,1}} - \nu^{(0,1)}_{{y}_{2,1}} & \lambda - \nu_{y_{2,1}} \; \mu & 0 & \lambda - \nu_{x_{2,1}} \; \mu & 0 & 0\\ 0 & \lambda + \nu_{x_{2,2}} \; \mu & 0 & \lambda + \nu_{y_{2,2}} \; \mu & \beta - \nu^{(1,0)}_{{x}_{2,2}} - \nu^{(0,1)}_{{y}_{2,2}} & \lambda - \nu_{y_{2,2}} \; \mu & 0 & \lambda - \nu_{x_{2,2}} \; \mu & 0\\ 0 & 0 & \lambda + \nu_{x_{2,3}} \; \mu & 0 & \lambda + \nu_{y_{2,3}} \; \mu & \beta - \nu^{(1,0)}_{{x}_{2,3}} - \nu^{(0,1)}_{{y}_{2,3}} & 0 & 0 & \lambda - \nu_{x_{2,3}} \; \mu\\ \hline\\ 0 & 0 & 0 & \lambda + \nu_{x_{3,1}} \; \mu & 0 & 0 & \beta - \nu^{(1,0)}_{{x}_{3,1}} - \nu^{(0,1)}_{{y}_{3,1}} & \lambda - \nu_{y_{3,1}} \; \mu & 0\\ 0 & 0 & 0 & 0 & \lambda + \nu_{x_{3,2}} \; \mu & 0 & \lambda + \nu_{y_{3,2}} \; \mu & \beta - \nu^{(1,0)}_{{x}_{3,2}} - \nu^{(0,1)}_{{y}_{3,2}} & \lambda - \nu_{y_{3,2}} \; \mu\\ 0 & 0 & 0 & 0 & 0 & \lambda + \nu_{x_{3,3}} \; \mu & 0 & \lambda + \nu_{y_{3,3}} \; \mu & \beta -\nu^{(1,0)}_{{x}_{3,3}} - \nu^{(0,1)}_{{y}_{3,3}} \end{array} \end{bmatrix}\begin{bmatrix} C^{k}_{1,1}\\C^{k}_{1,2}\\C^{k}_{1,3}\\\hline\\C^{k}_{2,1}\\C^{k}_{2,2}\\C^{k}_{2,3}\\\hline\\C^{k}_{3,1}\\C^{k}_{3,2}\\C^{k}_{3,3}\end{bmatrix}$$

3) Now, we wrote the matrixes in $\;n^2\;$ matrix blocks of $\;n\;$ x $\;n\;$ dimension, and the vectors in $\;n\;$ vector blocks of $\;n\;$ dimension:

$$ \underline{\underline{A}} \; = \begin{bmatrix} \underline{\underline{A_{1,1}}} & \underline{\underline{A_{1,2}}} & \cdots & \underline{\underline{A_{1,n}}}\\ \underline{\underline{A_{2,1}}} & \underline{\underline{A_{2,2}}} & \cdots & \underline{\underline{A_{2,n}}}\\ \vdots & \vdots & \ddots & \vdots \\ \underline{\underline{A_{n,1}}} & \underline{\underline{A_{n,2}}} & \cdots & \underline{\underline{A_{n,n}}}\\ \end{bmatrix} \;\;\;\; \underline{C^{k+1}} \; = \begin{bmatrix} \underline{C^{k+1}_{1}} \\ \underline{C^{k+1}_{2}} \\ \vdots \\ \underline{C^{k+1}_{n}} \\ \end{bmatrix} \;\;\;\; \underline{\underline{B}} \; = \begin{bmatrix} \underline{\underline{B_{1,1}}} & \underline{\underline{B_{1,2}}} & \cdots & \underline{\underline{B_{1,n}}}\\ \underline{\underline{B_{2,1}}} & \underline{\underline{B_{2,2}}} & \cdots & \underline{\underline{B_{2,n}}}\\ \vdots & \vdots & \ddots & \vdots \\ \underline{\underline{B_{n,1}}} & \underline{\underline{B_{n,2}}} & \cdots & \underline{\underline{B_{n,n}}}\\ \end{bmatrix} \;\;\;\; \underline{C^{k}} \; = \begin{bmatrix} \underline{C^{k}_{1}} \\ \underline{C^{k}_{2}} \\ \vdots \\ \underline{C^{k}_{n}} \\ \end{bmatrix} $$
And finally, we realised the pattern it followed, and could generalized for a $\;n\;$ x $\;n\;$ space steps:

$$ \underline{\underline{A_{w,w}}} \; = \begin{bmatrix} 1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{w,1}} + \nu^{(0,1)}_{{y}_{w,1}} & - \lambda + \nu_{y_{w,1}} \; \mu & 0 & \cdots & 0\\ - \lambda - \nu_{y_{w,2}} \; \mu & 1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{w,2}} + \nu^{(0,1)}_{{y}_{w,2}} & - \lambda + \nu_{y_{w,2}} \; \mu & \cdots & 0\\ 0 & - \lambda - \nu_{y_{w,3}} \; \mu & 1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{w,3}} + \nu^{(0,1)}_{{y}_{w,3}} & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & 1 + 4 \; \lambda + \nu^{(1,0)}_{{x}_{w,n}} + \nu^{(0,1)}_{{y}_{w,n}}\\ \end{bmatrix} $$ $$ \underline{\underline{A_{w,w-1}}} \; = \begin{bmatrix} - \lambda - \nu_{x_{w,1}} \; \mu & 0 & 0 & \cdots & 0\\ 0 & - \lambda - \nu_{x_{w,2}} \; \mu & 0 & \cdots & 0\\ 0 & 0 & - \lambda - \nu_{x_{w,3}} \; \mu & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & - \lambda - \nu_{x_{w,n}} \; \mu \\ \end{bmatrix} $$ $$ \underline{\underline{A_{w-1,w}}} \; = \begin{bmatrix} - \lambda + \nu_{x_{w-1,1}} \; \mu & 0 & 0 & \cdots & 0\\ 0 & - \lambda + \nu_{x_{w-1,2}} \; \mu & 0 & \cdots & 0\\ 0 & 0 & - \lambda + \nu_{x_{w-1,3}} \; \mu & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & - \lambda + \nu_{x_{w-1,n}} \; \mu \\ \end{bmatrix} $$
4) The last step focuses on isolating the unknowns vector, by doing:

$$ \underline{C^{k+1}} \; = \; \underline{\underline{A^{-1}}} \; \underline{\underline{B}} \; \underline{C^{k}} \; = \; \underline{\underline{\mathcal{M}}} \; \underline{C^{k}}$$
Where $\;\underline{\underline{\mathcal{M}}} \; = \; \underline{\underline{A^{-1}}} \; \underline{\underline{B}}$
That is the last ecuation system must be solved each time iteration. Note that, for the first iteration $\;(k\;=\;0)$, we get $\;\underline{C^{1}}\;=\;\underline{\underline{\mathcal{M}}} \; \underline{C^{0}}\;$, and this $\;\underline{C^{0}}\;$ vector, is given with the initial conditions, in our case, a Gaussian Distribution, centered in the middle of the plate.