#+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