#+TITLE: Numerical solution of diffusion equation in 2D with ADI Scheme
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* Finite differences with nodes as cells' centres

The 1D diffusion equation is:

#+NAME: eqn:1
\begin{align}
\frac{\partial C }{\partial t} & = \frac{\partial}{\partial x} \left(\alpha \frac{\partial C }{\partial x} \right) \nonumber \\
   & = \alpha \frac{\partial^2 C}{\partial x^2}
\end{align}

We aim at numerically solving [[eqn:1]] on a spatial grid such as:

[[./grid_pqc.pdf]]

The left boundary is defined on $x=0$ while the center of the first
cell - which are the points constituting the finite difference nodes -
is in $x=dx/2$, with $dx=L/n$.


** The explicit FTCS scheme (as in PHREEQC)

We start by discretizing [[eqn:1]] following an explicit Euler scheme and
specifically a Forward Time, Centered Space finite difference. 

For each cell index $i \in 1, \dots, n-1$ and assuming constant
$\alpha$, we can write:

#+NAME: eqn:2
\begin{equation}\displaystyle
   \frac{C_i^{j+1} -C_i^{j}}{\Delta t} = \alpha\frac{\frac{C^j_{i+1}-C^j_{i}}{\Delta x}-\frac{C^j_{i}-C^j_{i-1}}{\Delta x}}{\Delta x}
\end{equation}

In practice, we evaluate the first derivatives of $C$ w.r.t. $x$ on
the boundaries of each cell (i.e., $(C_{i+1}-C_i)/\Delta x$ on the
right boundary of the i-th cell and $(C_{i}-C_{i-1})/\Delta x$ on its
left cell boundary) and then repeat the differentiation to get the
second derivative of $C$ on the the cell centre $i$.

This discretization works for all internal cells, but not for the
domain boundaries ($i=0$ and $i=n$). To properly treat them, we need
to account for the discrepancy in the discretization.

For the first (left) cell, whose center is at $x=dx/2$, we can
evaluate the left gradient with the left boundary using such distance,
calling $l$ the numerical value of a constant boundary condition:

#+NAME: eqn:3
\begin{equation}\displaystyle
\frac{C_0^{j+1} -C_0^{j}}{\Delta t} = \alpha\frac{\frac{C^j_{1}-C^j_{0}}{\Delta x}-
\frac{C^j_{0}-l}{\frac{\Delta x}{2}}}{\Delta x}
\end{equation}

This expression, once developed, yields:

#+NAME: eqn:4
\begin{align}\displaystyle
C_0^{j+1} & =  C_0^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( C^j_{1}-C^j_{0}- 2 C^j_{0}+2l \right) \nonumber \\
          & =  C_0^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( C^j_{1}- 3 C^j_{0} +2l \right)
\end{align}


In case of constant right boundary, the finite difference of point
$C_n$ - calling $r$ the right boundary value - is:

#+NAME: eqn:5
\begin{equation}\displaystyle
\frac{C_n^{j+1} -C_n^j}{\Delta t} = \alpha\frac{\frac{r - C^j_{n}}{\frac{\Delta x}{2}}-
\frac{C^j_{n}-C^j_{n-1}}{\Delta x}}{\Delta x}
\end{equation}

Which, developed, gives
#+NAME: eqn:6
\begin{align}\displaystyle
C_n^{j+1} & =  C_n^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( 2 r - 2 C^j_{n} -C^j_{n} + C^j_{n-1} \right) \nonumber \\
          & =  C_n^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( 2 r - 3 C^j_{n} + C^j_{n-1} \right)
\end{align}

If on the right boundary we have closed or Neumann condition, the left derivative in eq. [[eqn:5]]
becomes zero and we are left with:


#+NAME: eqn:7
\begin{equation}\displaystyle
C_n^{j+1} = C_n^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot (C^j_{n-1} - C^j_n)
\end{equation}



A similar treatment can be applied to the BTCS implicit scheme.

** Implicit BTCS scheme

First, we define the Backward Time difference:

\begin{equation}
    \frac{\partial C^{j+1} }{\partial t} = \frac{C^{j+1}_i - C^{j}_i}{\Delta t}
\end{equation}

Second the spatial derivative approximation, evaluated at time level $j+1$:

\begin{equation}
    \frac{\partial^2 C^{j+1} }{\partial x^2} = \frac{\frac{C^{j+1}_{i+1}-C^{j+1}_{i}}{\Delta x}-\frac{C^{j+1}_{i}-C^{j+1}_{i-1}}{\Delta x}}{\Delta x}
\end{equation}

Taking the 1D diffusion equation from [[eqn:1]] and substituting each term by the
equations given above leads to the following equation:


# \begin{equation}\displaystyle
#    \frac{C_i^{j+1} -C_i^{j}}{\Delta t} = \alpha\frac{\frac{C^{j+1}_{i+1}-C^{j+1}_{i}}{\Delta x}-\frac{C^{j+1}_{i}-C^{j+1}_{i-1}}{\Delta x}}{\Delta x}
# \end{equation}

# Since we are not able to solve this system w.r.t unknown values in $C^{j-1}$ we
# are shifting each j by 1 to $j \to (j+1)$ and $(j-1) \to j$ which leads to:

\begin{align}\displaystyle
\frac{C_i^{j+1} - C_i^{j}}{\Delta t}    & = \alpha\frac{\frac{C^{j+1}_{i+1}-C^{j+1}_{i}}{\Delta x}-\frac{C^{j+1}_{i}-C^{j+1}_{i-1}}{\Delta x}}{\Delta x} \nonumber \\
                                        & = \alpha\frac{C^{j+1}_{i-1} - 2C^{j+1}_{i} + C^{j+1}_{i+1}}{\Delta x^2}
\end{align}

This only applies to inlet cells with no ghost node as neighbor. For the left
cell with its center at $\frac{dx}{2}$ and the constant concentration on the
left ghost node called $l$ the equation goes as followed:

\begin{equation}\displaystyle
\frac{C_0^{j+1} -C_0^{j}}{\Delta t} = \alpha\frac{\frac{C^{j+1}_{1}-C^{j+1}_{0}}{\Delta x}-
\frac{C^{j+1}_{0}-l}{\frac{\Delta x}{2}}}{\Delta x}
\end{equation}

This expression, once developed, yields:

\begin{align}\displaystyle
C_0^{j+1} & =  C_0^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( C^{j+1}_{1}-C^{j+1}_{0}- 2 C^{j+1}_{0}+2l \right) \nonumber \\
          & =  C_0^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( C^{j+1}_{1}- 3 C^{j+1}_{0} +2l \right)
\end{align}

Now we define variable $s_x$ as followed:

\begin{equation}
    s_x = \frac{\alpha \cdot \Delta t}{\Delta x^2}
\end{equation}

Substituting with the new variable $s_x$ and reordering of terms leads to the equation applicable to our model:

\begin{equation}\displaystyle
    -C^j_0 = s_x \cdot C^{j+1}_1 + (2s_x) \cdot l + (-1 - 3s_x) \cdot C^{j+1}_0
\end{equation}

*TODO*
- Right boundary
- Tridiagonal matrix filling




#+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

$b(i,j) = \begin{cases}
bc(i,j).\text{value} & \text{if } bc(i,N-1) = \text{constant} \\
-c(i,j)-p(i,j) & \text{else}
\end{cases}$