doc: remove old stuff from ADI documentation

doc/ADI_scheme.org

@@ -305,104 +305,3 @@ form:
 
 
 #+LATEX: \clearpage
-
-* Old stuff
-
-** 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
-
-$p(i,j) = \frac{\Delta t}{2}\alpha(i,j)\frac{c(i-1,j) - 2\cdot c(i,j) + c(i+1,j)}{\Delta x^2}$
-
-\noindent $p$ is called =t0_c= inside code
-
-$b(i,j) = \begin{cases}
-bc(i,j).\text{value} & \text{if } bc(i,N-1) = \text{constant} \\
--c(i,j)-p(i,j) & \text{else}
-\end{cases}$