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 + V_{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 + V_{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 + V_{1,3} & 0 & 0 & - \lambda + \nu_{x_{1,3}} \; \mu & 0 & 0 & 0\\ \hline\\ - \lambda - \nu_{x_{2,1}} \; \mu & 0 & 0 & \alpha + V_{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 + V_{2,1} & - \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 + V_{2,3} & 0 & 0 & - \lambda + \nu_{x_{2,3}} \; \mu\\ \hline\\ 0 & 0 & 0 & - \lambda - \nu_{x_{3,1}} \; \mu & 0 & 0 & \alpha + V_{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 + V_{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 + V_{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}$$=$

$\begin{bmatrix} \begin{array}{ccc|ccc|ccc} \beta - V_{1,1} & \lambda - \nu_{y_{1,1}} \; \mu & 0 & \lambda - \nu_{x_{1,1}} \; \mu & 0 & 0 & 0 & 0 & 0\\ \lambda + \nu_{y_{1,2}} \; \mu & \beta - V_{1,2} & \lambda - \nu_{y_{1,2}} \; \mu & 0 & \lambda - \nu_{x_{1,2}} \; \mu & 0 & 0 & 0 & 0\\ 0 & \lambda + \nu_{y_{1,3}} \; \mu & \beta - V_{1,3} & 0 & 0 & \lambda - \nu_{x_{1,3}} \; \mu & 0 & 0 & 0\\ \hline\\ \lambda + \nu_{x_{2,1}} \; \mu & 0 & 0 & \beta - V_{2,1} & \lambda - \nu_{y_{2,1}} \; \mu & 0 & \lambda - \nu_{x_{2,1}} \; \mu & 0 & 0\\ 0 & \lambda + \nu_{x_{2,2}} \; \mu & 0 & \lambda + \nu_{y_{2,2}} \; \mu & \beta - V_{2,2} & \lambda - \nu_{y_{2,2}} \; \mu & 0 & \lambda - \nu_{x_{2,2}} \; \mu & 0\\ 0 & 0 & \lambda + \nu_{x_{2,3}} \; \mu & 0 & \lambda + \nu_{y_{2,3}} \; \mu & \beta - V_{2,3} & 0 & 0 & \lambda - \nu_{x_{2,3}} \; \mu\\ \hline\\ 0 & 0 & 0 & \lambda + \nu_{x_{3,1}} \; \mu & 0 & 0 & \beta - V_{3,1} & \lambda - \nu_{y_{3,1}} \; \mu & 0\\ 0 & 0 & 0 & 0 & \lambda + \nu_{x_{3,2}} \; \mu & 0 & \lambda + \nu_{y_{3,2}} \; \mu & \beta - V_{3,2} & \lambda - \nu_{y_{3,2}} \; \mu\\ 0 & 0 & 0 & 0 & 0 & \lambda + \nu_{x_{3,3}} \; \mu & 0 & \lambda + \nu_{y_{3,3}} \; \mu & \beta -V_{3,3} \end{array} \end{bmatrix}\begin{bmatrix} C^{k}_{1,1}\\C^{k}_{1,2}\\C^{k}_{1,3}\\\hline\\C^{k}_{2,1}\\C^{k}_{2,2}\\C^{k}_{2,3}\\\hline\\C^{k}_{3,1}\\C^{k}_{3,2}\\C^{k}_{3,3}\end{bmatrix}$

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.