Added org document of filling scheme

doc/.gitignore

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+/doc/*.pdf
+/doc/*.tex

doc/ADI_scheme.org

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+#+TITLE: Adi Scheme
+
+* Input
+
+- =c= $\rightarrow c$
+  - containing current concentrations at each grid cell for species
+  - size: $N \times M$
+  - row-major
+- =alpha= $\rightarrow \alpha$
+  - diffusion coefficient for both directions (x and y)
+  - size: $N \times M$
+  - row-major
+- =boundary_condition= $\rightarrow bc$
+  - Defines closed or constant boundary condition for each grid cell
+  - size: $N \times M$
+  - row-major
+
+* Internals
+
+- =A_matrix= $\rightarrow A$
+  - coefficient matrix for linear equation system implemented as sparse matrix
+  - size: $((N+2)\cdot M) \times ((N+2)\cdot M)$ (including ghost zones in x direction)
+  - column-major (not relevant)
+
+- =b_vector= $\rightarrow b$
+  - right hand side of the linear equation system
+  - size: $(N+2) \cdot M$
+  - column-major (not relevant)
+- =x_vector= $\rightarrow x$
+  - solutions of the linear equation system
+  - size: $(N+2) \cdot M$
+  - column-major (not relevant)
+
+* Calculation for $\frac{1}{2}$ timestep
+
+** Symbolic addressing of grid cells
+[[./grid.png]]
+
+** Filling of matrix $A$
+
+- row-wise iterating with $i$ over =c= and =\alpha= matrix respectively
+- addressing each element of a row with $j$
+- matrix $A$ also containing $+2$ ghost nodes for each row of input matrix $\alpha$
+  - $\rightarrow offset = N+2$
+  - addressing each object $(i,j)$ in matrix $A$ with $(offset \cdot i + j, offset \cdot i + j)$
+
+*** Rules
+
+$s_x(i,j) = \frac{\alpha(i,j)*\frac{t}{2}}{\Delta x^2}$ where $x$ defining the domain size in x direction.
+
+For the sake of simplicity we assume that each row of the $A$ matrix is addressed correctly with the given offset.
+
+**** Ghost nodes
+
+$A(i,-1) = 1$
+
+$A(i,N) = 1$
+
+**** Inlet
+
+$A(i,j) = \begin{cases}
+1 & \text{if } bc(i,j) = \text{constant} \\
+-1-2*s_x(i,j) & \text{else}
+\end{cases}$
+
+$A(i,j\pm 1) = \begin{cases}
+0 & \text{if } bc(i,j) = \text{constant} \\
+s_x(i,j) & \text{else}
+\end{cases}$
+
+** Filling of vector $b$
+
+- each elements assign a concrete value to the according value of the row of matrix $A$
+- Adressing would look like this: $(i,j) = b(i \cdot (N+2) + j)$
+  - $\rightarrow$ for simplicity we will write $b(i,j)$
+
+
+
+
+*** Rules
+
+**** Ghost nodes
+
+$b(i,-1) = \begin{cases}
+0 & \text{if } bc(i,0) = \text{constant} \\
+c(i,0) & \text{else}
+\end{cases}$
+
+$b(i,N) = \begin{cases}
+0 & \text{if } bc(i,N-1) = \text{constant} \\
+c(i,N-1) & \text{else}
+\end{cases}$
+
+*** Inlet
+
+$p(i,j) = \alpha(i,j)\frac{c(i-1,j) - 2\cdot c(i,j) + c(i+1,j)}{\Delta x^2}$[fn:1]
+
+$b(i,j) = \begin{cases}
+bc(i,j).\text{value} & \text{if } bc(i,N-1) = \text{constant} \\
+-c(i,j)-p(i,j) & \text{else}
+\end{cases}$
+
+[fn:1] $p$ is called =t0_c= inside code