diff --git a/doc/ADI_scheme.org b/doc/ADI_scheme.org index 50544fe..654d5e8 100644 --- a/doc/ADI_scheme.org +++ b/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 - -$b(i,j) = \begin{cases} -bc(i,j).\text{value} & \text{if } bc(i,N-1) = \text{constant} \\ --c(i,j)-p(i,j) & \text{else} -\end{cases}$ diff --git a/doc/grid.png b/doc/grid.png deleted file mode 100644 index 86940a4..0000000 Binary files a/doc/grid.png and /dev/null differ