#+TITLE: Finite Difference Schemes for the numerical solution of heterogeneous diffusion equation in 2D
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* Homogeneous diffusion in 1D

** Finite differences with nodes as cells' centres

The 1D diffusion equation for spatially constant diffusion
coefficients $\alpha$ 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:

[[./images/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^{t+1} -C_i^{t}}{\Delta t} = \alpha\frac{\frac{C^t_{i+1}-C^t_{i}}{\Delta x}-\frac{C^t_{i}-C^t_{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^{t+1} -C_0^{t}}{\Delta t} = \alpha\frac{\frac{C^t_{1}-C^t_{0}}{\Delta x}-
\frac{C^t_{0}-l}{\frac{\Delta x}{2}}}{\Delta x}
\end{equation}

This expression, once developed, yields:

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

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

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

#+NAME: eqn:secondderivative
\begin{equation}
    \frac{\partial^2 C^{t+1} }{\partial x^2} = \frac{\frac{C^{t+1}_{i+1}-C^{t+1}_{i}}{\Delta x}-\frac{C^{t+1}_{i}-C^{t+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:

#+NAME: eqn:1DBTCS
\begin{align}\displaystyle
\frac{C_i^{t+1} - C_i^{t}}{\Delta t}    & = \alpha\frac{\frac{C^{t+1}_{i+1}-C^{t+1}_{i}}{\Delta x}-\frac{C^{t+1}_{i}-C^{t+1}_{i-1}}{\Delta x}}{\Delta x} \nonumber \\
                                        & = \alpha\frac{C^{t+1}_{i-1} - 2C^{t+1}_{i} + C^{t+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^{t+1} -C_0^{t}}{\Delta t} = \alpha\frac{\frac{C^{t+1}_{1}-C^{t+1}_{0}}{\Delta x}-
\frac{C^{t+1}_{0}-l}{\frac{\Delta x}{2}}}{\Delta x}
\end{equation}

This expression, once developed, yields:

\begin{align}\displaystyle
C_0^{t+1} & =  C_0^{t} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( C^{t+1}_{1}-C^{t+1}_{0}- 2 C^{t+1}_{0}+2l \right) \nonumber \\
          & =  C_0^{t} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( C^{t+1}_{1}- 3 C^{t+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^t_0 = (2s_x) \cdot l + (-1 - 3s_x) \cdot C^{t+1}_0 + s_x \cdot C^{t+1}_1
\end{equation}

The right boundary follows the same scheme. We now want to show the equation for the rightmost inlet cell $C_n$ with right boundary value $r$:

\begin{equation}\displaystyle
\frac{C_n^{t+1} -C_n^{t}}{\Delta t} = \alpha\frac{\frac{r-C^{t+1}_{n}}{\frac{\Delta x}{2}}-
\frac{C^{t+1}_{n}-C^{t+1}_{n-1}}{\Delta x}}{\Delta x}
\end{equation}

This expression, once developed, yields:

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

Now rearrange terms and substituting with $s_x$ leads to:

\begin{equation}\displaystyle
    -C^t_n = s_x \cdot C^{t+1}_{n-1} + (-1 - 3s_x) \cdot C^{t+1}_n + (2s_x) \cdot r
\end{equation}

*TODO*
- Tridiagonal matrix filling

#+LATEX: \clearpage

* Diffusion in 2D: the Alternating Direction Implicit scheme


In 2D, the diffusion equation (in absence of source terms) and
assuming homogeneous but anisotropic diffusion coefficient
$\alpha=(\alpha_x,\alpha_y)$ becomes:

#+NAME: eqn:2d
\begin{equation}
\displaystyle  \frac{\partial C}{\partial t} = \alpha_x \frac{\partial^2 C}{\partial x^2} + \alpha_y\frac{\partial^2 C}{\partial y^2}
\end{equation}

** 2D ADI using BTCS scheme

The Alternating Direction Implicit method consists in splitting the
integration of eq. [[eqn:2d]] in two half-steps, each of which represents
implicitly the derivatives in one direction, and explicitly in the
other. Therefore we make use of second derivative operator defined in
equation [[eqn:secondderivative]] in both $x$ and $y$ direction for half
the time step $\Delta t$.

Denoting $i$ the grid cell index along $x$ direction, $j$ the index in
$y$ direction and $t$ as the time level, the spatially centered second
derivatives can be written as:

\begin{align}\displaystyle
\frac{\partial^2 C^t_{i,j}}{\partial x^2} &= \frac{C^{t}_{i-1,j} - 2C^{t}_{i,j} + C^{t}_{i+1,j}}{\Delta x^2} \\
\frac{\partial^2 C^t_{i,j}}{\partial y^2} &= \frac{C^{t}_{i,j-1} - 2C^{t}_{i,j} + C^{t}_{i,j+1}}{\Delta y^2}
\end{align}

The ADI scheme is formally defined by the equations:

#+NAME: eqn:genADI
\begin{equation}
\systeme{ 
  \displaystyle  \frac{C^{t+1/2}_{i,j}-C^t_{i,j}}{\Delta t/2}     = \displaystyle \alpha_x \frac{\partial^2 C^{t+1/2}_{i,j}}{\partial x^2} + \alpha_y \frac{\partial^2 C^{t}_{i,j}}{\partial y^2},
  \displaystyle  \frac{C^{t+1}_{i,j}-C^{t+1/2}_{i,j}}{\Delta t/2} = \displaystyle \alpha_x \frac{\partial^2 C^{t+1/2}_{i,j}}{\partial x^2} + \alpha_y \frac{\partial^2 C^{t+1}_{i,j}}{\partial y^2}
}
\end{equation}

\noindent The first of equations [[eqn:genADI]], which writes implicitly
the spatial derivatives in $x$ direction, after bringing the $\Delta t
/ 2$ terms on the right hand side and substituting $s_x=\frac{\alpha_x
\cdot \Delta t}{2 \Delta x^2}$ and $s_y=\frac{\alpha_y\cdot\Delta
t}{2\Delta y^2}$ reads:

\begin{equation}\displaystyle
C^{t+1/2}_{i,j}-C^t_{i,j} = s_x (C^{t+1/2}_{i-1,j} - 2C^{t+1/2}_{i,j} + C^{t+1/2}_{i+1,j}) + s_y (C^{t}_{i,j-1} - 2C^{t}_{i,j} + C^{t}_{i,j+1})
\end{equation}

\noindent Separating the known terms (at time level $t$) on the left
hand side and the implicit terms (at time level $t+1/2$) on the right
hand side, we get:

#+NAME: eqn:sweepX
\begin{equation}\displaystyle
-C^t_{i,j} - s_y (C^{t}_{i,j-1} - 2C^{t}_{i,j} + C^{t}_{i,j+1}) = - C^{t+1/2}_{i,j} + s_x (C^{t+1/2}_{i-1,j} - 2C^{t+1/2}_{i,j} + C^{t+1/2}_{i+1,j}) 
\end{equation}

\noindent Equation [[eqn:sweepX]] can be solved with a BTCS scheme since
it corresponds to a tridiagonal system of equations, and resolves the
$C^{t+1/2}$ at each inner grid cell.

The second of equations [[eqn:genADI]] can be treated the same way and
yields:

#+NAME: eqn:sweepY
\begin{equation}\displaystyle
-C^{t + 1/2}_{i,j} - s_x (C^{t + 1/2}_{i-1,j} - 2C^{t + 1/2}_{i,j} + C^{t + 1/2}_{i+1,j}) = - C^{t+1}_{i,j} + s_y (C^{t+1}_{i,j-1} - 2C^{t+1}_{i,j} + C^{t+1}_{i,j+1}) 
\end{equation}

This scheme only applies to inner cells, or else $\forall i,j \in [1,
n-1] \times [1, n-1]$. Following an analogous treatment as for the 1D
case, and noting $l_x$ and $l_y$ the constant left boundary values and
$r_x$ and $r_y$ the right ones for each direction $x$ and $y$, we can
modify equations [[eqn:sweepX]] for $i=0, j \in [1, n-1]$

#+NAME: eqn:boundXleft
\begin{equation}\displaystyle
-C^t_{0,j} - s_y (C^{t}_{0,j-1} - 2C^{t}_{0,j} + C^{t}_{0,j+1}) = - C^{t+1/2}_{0,j} + s_x (C^{t+1/2}_{1,j} - 3C^{t+1/2}_{0,j} + 2 l_x) 
\end{equation}

\noindent Similarly for  $i=n, j \in [1, n-1]$:
#+NAME: eqn:boundXright
\begin{equation}\displaystyle
-C^t_{n,j} - s_y (C^{t}_{n,j-1} - 2C^{t}_{n,j} + C^{t}_{n,j+1}) = - C^{t+1/2}_{n,j} + s_x (C^{t+1/2}_{n-1,j} - 3C^{t+1/2}_{n,j} + 2 r_x) 
\end{equation}

\noindent For $i=j=0$:
#+NAME: eqn:bound00
\begin{equation}\displaystyle
-C^t_{0,0} - s_y (C^{t}_{0,1} - 3C^{t}_{0,0} + 2l_y) = - C^{t+1/2}_{0,0} + s_x (C^{t+1/2}_{1,0} - 3C^{t+1/2}_{0,0} + 2 l_x) 
\end{equation}

Analogous expressions are readily derived for all possible
combinations of $i,j \in 0\times n$. In practice, wherever an index
$i$ or $j$ is $0$ or $n$, the centered spatial derivatives in $x$ or
$y$ directions must be substituted in relevant parts of the sweeping
equations \textbf{in both the implicit or the explicit sides} of
equations [[eqn:sweepX]] and [[eqn:sweepY]] by a term

#+NAME: eqn:bound00
\begin{equation}\displaystyle
 s(C_{forw} - 3C + 2 bc) 
\end{equation}
\noindent where $bc$ is the boundary condition in the given direction,
$s$ is either $s_x$ or $s_y$, and $C_{forw}$ indicates the contiguous
cell opposite to the boundary. Alternatively, noting the second
derivative operator as $\partial_{dir}^2$, we can write in compact
form:

\begin{equation}
\systeme{
  \displaystyle  \partial_x^2 C_{0,j} = 2l_x - 3C_{0,j} + C_{1,j} ,
  \displaystyle  \partial_x^2 C_{n,j} = 2r_x - 3C_{n,j} + C_{n-1,j} ,
  \displaystyle  \partial_y^2 C_{i,0} = 2l_y - 3C_{i,0} + C_{i,1} ,
  \displaystyle  \partial_y^2 C_{i,n} = 2r_y - 3C_{i,n} + C_{i,n-1}
}
\end{equation}



#+LATEX: \clearpage

* Heterogeneous diffusion

If the diffusion coefficient $\alpha$ is spatially variable, equation
[[eqn:1]] can be rewritten as:

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

** Discretization of the equation using chain rule

\noindent From the product rule for derivatives we obtain:

#+NAME: eqn:product
\begin{equation}
\frac{\partial}{\partial x} \left(\alpha(x) \frac{\partial C }{\partial x}\right) = \frac{\partial \alpha}{\partial x} \cdot \frac{\partial C}{\partial x} + \alpha \frac{\partial^2 C }{\partial x^2} 
\end{equation}

\noindent Using a spatially centred second order finite difference
approximation at $x=x_i$ for both $\alpha$ and $C$, we have
#+NAME: eqn:hetdiff_fd
\begin{align}
\frac{\partial \alpha}{\partial x} \cdot \frac{\partial C}{\partial x} +
 \alpha \frac{\partial^2 C }{\partial x^2} & \simeq \frac{\alpha_{i+1} - \alpha_{i-1}}{2\Delta x}\cdot\frac{C_{i+1} - C_{i-1}}{2\Delta x} + \alpha_i \frac{C_{i+1} - 2 C_{i} + C_{i-1}}{\Delta x^2} \\ \nonumber
 & = \frac{1}{\Delta x^2} \frac{\alpha_{i+1} - \alpha_{i-1}}{4}(C_{i+1} - C_{i-1}) + \frac{\alpha_{i}}{\Delta x^2}(C_{i+1}-2C_i+C_{i-1})\\ \nonumber
 & = \frac{1}{\Delta x^2} \left\{A C_{i+1} -2\alpha_i C_i + AC_{i-1})\right\}
\end{align}
\noindent having set
\[ A = \frac{\alpha_{i+1}-\alpha_{i-1}}{4} + \alpha_i \]

\noindent In 2D the ADI scheme [[eqn:genADI]] with heterogeneous diffusion
coefficients can thus be written:

#+NAME: eqn:genADI_het
\begin{equation}
\systeme{ 
  \displaystyle \frac{C^{t+1/2}_{i,j}-C^{t    }_{i,j}}{\Delta t/2} = \displaystyle \frac{\partial}{\partial x} \left( \alpha^x_{i,j} \frac{\partial C^{t+1/2}_{i,j}}{\partial x}\right) + \frac{\partial}{\partial y} \left( \alpha^y_{i,j} \frac{\partial C^{t  }_{i,j}}{\partial y}\right),
  \displaystyle \frac{C^{t+1  }_{i,j}-C^{t+1/2}_{i,j}}{\Delta t/2} = \displaystyle \frac{\partial}{\partial x} \left( \alpha^x_{i,j} \frac{\partial C^{t+1/2}_{i,j}}{\partial x}\right) + \frac{\partial}{\partial y} \left( \alpha^y_{i,j} \frac{\partial C^{t+1}_{i,j}}{\partial y}\right)
}
\end{equation}

\noindent We define for compactness $S_x=\frac{\Delta t}{2\Delta x^2}$
and $S_y=\frac{\Delta t}{2\Delta y^2}$ and

#+NAME: eqn:het_AB
\begin{equation}
\systeme{ 
  \displaystyle  A_{i,j} = \displaystyle \frac{\alpha^x_{i+1,j}  -\alpha^x_{i-1,j  }}{4} + \alpha^x_{i,j},
  \displaystyle  B_{i,j} = \displaystyle \frac{\alpha^y_{i,  j+1}-\alpha^y_{i  ,j-1}}{4} + \alpha^y_{i,j}
}
\end{equation}

\noindent Plugging eq. ([[eqn:hetdiff_fd]]) into the first of equations
([[eqn:genADI_het]]) - so called "sweep by x" - and putting all implicit
terms (at time level $t+1/2$) on the left hand side we obtain:

#+NAME: eqn:sweepX_het
\begin{equation}\displaystyle
\begin{split}
-S_x A_{i,j} C^{t+1/2}_{i+1,j} + (1 + 2S_x\alpha^x_{i,j})C^{t+1/2}_{i,j}  - S_x A_{i,j}C^{t+1/2}_{i-1,j} = \\
 S_y B_{i,j} C^{t    }_{i,j+1} + (1 - 2S_y\alpha^y_{i,j})C^{t    }_{i,j}  + S_y B_{i,j}C^{t    }_{i,j-1}
\end{split}
\end{equation}

\noindent In the same way for the second of eq. [[eqn:genADI_het]] we have:

#+NAME: eqn:sweepY_het
\begin{equation}\displaystyle
\begin{split}
-S_y B_{i,j} C^{t+1  }_{i,j+1} + (1 + 2S_y\alpha^y_{i,j})C^{t+1  }_{i,j}  - S_y B_{i,j}C^{t+1  }_{i,j-1} = \\
 S_x A_{i,j} C^{t+1/2}_{i+1,j} + (1 - 2S_x\alpha^x_{i,j})C^{t+1/2}_{i,j}  + S_x A_{i,j}C^{t+1/2}_{i-1,j}
\end{split}
\end{equation}

\noindent If the diffusion coefficients are constant,
$A_{i,j}=B_{i,j}=\alpha$ and the scheme reverts to the homogeneous
case. Problem with this discretization is that the terms in $A_{ij}$
and $B_{ij}$ can be negative depending on the derivative of the
diffusion coefficient, resulting in unphysical values for the
concentrations.

** Direct discretization

As noted in literature (LeVeque and Numerical Recipes) a better way is
to discretize directly the physical problem (eq. [[eqn:hetdiff]]) at
points halfway between grid points:

\begin{align*}
\begin{cases}
\displaystyle \alpha(x_{i+1/2}) \frac{\partial C }{\partial x}(x_{i+1/2}) & \displaystyle = \alpha_{i+1/2} \left( \frac{C_{i+1} -C_{i}}{\Delta x} \right) \\
\displaystyle \alpha(x_{i-1/2}) \frac{\partial C }{\partial x}(x_{i-1/2}) & \displaystyle = \alpha_{i-1/2} \left( \frac{C_{i} -C_{i-1}}{\Delta x} \right) 
\end{cases}
\end{align*}

\noindent A further differentiation gives us the spatially centered
approximation of $\frac{\partial}{\partial x} \left(\alpha(x)
\frac{\partial C }{\partial x}\right)$:

#+NAME: eqn:CS_het
\begin{equation}
\begin{aligned}
\frac{\partial}{\partial x} \left(\alpha(x)
\frac{\partial C }{\partial x}\right)(x_i) & \simeq \frac{1}{\Delta x}\left[\alpha_{i+1/2} \left( \frac{C_{i+1} -C_{i}}{\Delta x} \right) - \alpha_{i-1/2} \left( \frac{C_{i} -C_{i-1}}{\Delta x} \right) \right]\\
&\displaystyle =\frac{1}{\Delta x^2} \left[ \alpha_{i+1/2}C_{i+1} - (\alpha_{i+1/2}+\alpha_{i-1/2}) C_{i} + \alpha_{i-1/2}C_{i-1}\right]
\end{aligned}
\end{equation}

\noindent The ADI scheme with this approach becomes:

#+NAME: eqn:genADI_hetdir
\begin{equation}
\left\{
\begin{aligned}
   \frac{C^{t+1/2}_{i,j}-C^{t    }_{i,j}}{\Delta t/2} = & \frac{1}{\Delta x^2} \left[ \alpha_{i+1/2,j} C^{t+1/2}_{i+1,j} - (\alpha_{i+1/2,j}+\alpha_{i-1/2,j}) C^{t+1/2}_{i,j} + \alpha_{i-1/2,j} C^{t+1/2}_{i-1,j}\right] + \\
   & \frac{1}{\Delta y^2} \left[ \alpha_{i,j+1/2}C^{t}_{i,j+1} - (\alpha_{i,j+1/2}+\alpha_{i,j-1/2}) C^t_{i,j} + \alpha_{i,j-1/2}C^{t}_{i,j-1}\right]\\ 
\frac{C^{t+1  }_{i,j}-C^{t+1/2}_{i,j}}{\Delta t/2} = &  \frac{1}{\Delta y^2} \left[ \alpha_{i+1/2,j}C^{t+1/2}_{i+1,j} - (\alpha_{i+1/2,j}+\alpha_{i-1/2,j}) C^t_{i,j} + \alpha_{i-1/2,j}C^{t+1/2}_{i-1,j}\right] + \\
   & \frac{1}{\Delta x^2} \left[ \alpha_{i,j+1/2}C^{t+1}_{i,j+1} - (\alpha_{i,j+1/2}+\alpha_{i,j-1/2}) C^{t+1}_{i,j} + \alpha_{i,j-1/2}C^{t+1}_{i,j-1}\right] 
\end{aligned}
\right.
\end{equation}

\noindent Doing the usual algebra and separating implicit from
explicit terms, the two sweeps become:

#+NAME: eqn:sweepX_hetdir
\begin{equation}
\left\{
\begin{aligned}
-S_x \alpha^x_{i+1/2,j} C^{t+1/2}_{i+1,j} + (1 + S_x(\alpha^x_{i+1/2,j}+ \alpha^x_{i-1/2,j}))C^{t+1/2}_{i,j}  - S_x \alpha^x_{i-1/2,j} C^{t+1/2}_{i-1,j} = \quad & \\
 S_y \alpha^y_{i,j+1/2} C^{t    }_{i,j+1} + (1 - S_y(\alpha^y_{i,j+1/2}+ \alpha^y_{i,j-1/2}))C^{t    }_{i,j}  + S_y \alpha^y_{i,j-1/2} C^{t    }_{i,j-1} & \\[1em]
-S_y \alpha^y_{i,j+1/2} C^{t+1  }_{i,j+1} + (1 + S_y(\alpha^y_{i,j+1/2}+ \alpha^y_{i,j-1/2}))C^{t+1  }_{i,j}  - S_y \alpha^y_{i,j-1/2} C^{t+1  }_{i,j-1} = \qquad & \\
 S_x \alpha^x_{i+1/2,j} C^{t+1/2}_{i+1,j} + (1 - S_x(\alpha^x_{i+1/2,j}+ \alpha^x_{i-1/2,j}))C^{t+1/2}_{i,j}  + S_x \alpha^x_{i-1/2,j} C^{t+1/2}_{i-1,j} 
\end{aligned}
\right.
\end{equation}

\bigskip

\noindent The "interblock" diffusion coefficients $\alpha_{i+1/2,j}$
can be arithmetic mean:

\[
\displaystyle \alpha_{i+1/2, j} = \displaystyle \frac{\alpha_{i+1, j} + \alpha_{i, j}}{2}
\]

\noindent or the harmonic mean:

\[
\displaystyle \alpha_{i+1/2, j} = \displaystyle \frac{2}{\frac{1}{\alpha_{i+1, j}} + \frac{1}{\alpha_{i, j}}}
\]



\pagebreak

* Explicit scheme for 2D heterogeneous diffusion

A classical explicit FTCS scheme (forward in time, central in space)
for 2D heterogeneous diffusion can be expressed simply leveraging the
discretization of equation [[eqn:CS_het]]:

#+NAME: eqn:2DHeterFTCS
\begin{equation}
\begin{aligned}
\frac{C_{i,j}^{t+1} - C_{i,j}^{t}}{\Delta t} = & \frac{1}{\Delta x^2} \left[ \alpha^x_{i+1/2, j}C^t_{i+1, j} - (\alpha^x_{i+1/2, j} + \alpha^x_{i-1/2, j}) C^t_{i,j} + \alpha^x_{i-1/2,j}C^t_{i-1,j}\right] + \\
                                               & \frac{1}{\Delta y^2} \left[ \alpha^y_{i, j+1/2}C^t_{i, j+1} - (\alpha^y_{i, j+1/2} + \alpha^y_{i, j-1/2}) C^t_{i,j} + \alpha^y_{i,j-1/2}C^t_{i,j-1}\right]
\end{aligned}
\end{equation}
\noindent where in the RHS only the known concentrations at time $t$
appear. Rearranging the terms, we get:

#+NAME: eqn:2DHeterFTCS_final
\begin{equation}
\begin{aligned}
C_{i,j}^{t+1} = & C^t_{i,j} +\\
                & \frac{\Delta t}{\Delta x^2} \left[ \alpha^x_{i+1/2, j}C^t_{i+1, j} - (\alpha^x_{i+1/2, j} + \alpha^x_{i-1/2, j}) C^t_{i,j} + \alpha^x_{i-1/2,j}C^t_{i-1,j}\right] + \\
                & \frac{\Delta t}{\Delta y^2} \left[ \alpha^y_{i, j+1/2}C^t_{i, j+1} - (\alpha^y_{i, j+1/2} + \alpha^y_{i, j-1/2}) C^t_{i,j} + \alpha^y_{i,j-1/2}C^t_{i,j-1}\right]
\end{aligned}
\end{equation}

The Courant-Friedrichs-Lewy stability criterion (cfr Lee, 2017) for
this scheme reads:

#+NAME: eqn:CFL2DFTCS_Lee
\begin{equation}
\Delta t \leq \frac{1}{2 \max(\alpha_{i,j})} \cdot \frac{1}{\frac{1}{\Delta x^2} + \frac{1}{\Delta y^2}}
\end{equation}


Note that other derivations for the CFL condition are found in
literature. For example, the sources cited by [[https://en.wikipedia.org/wiki/FTCS_scheme][Wikipedia solution]] give:

#+NAME: eqn:CFL2DFTCS_wiki
\begin{equation}
\displaystyle \Delta t\leq {\frac {1}{4 \max(\alpha) \left({\frac {1}{\Delta x^{2}}}+{\frac {1}{\Delta y^{2}}}\right)}}
\end{equation}

We can produce a more restrictive condition than equation
[[eqn:CFL2DFTCS_Lee]] by considering the min of the $\Delta x$ and $\Delta
y$:

#+NAME: eqn:CFL2DFTCS
\begin{equation}
\Delta t \leq \frac{\min(\Delta x, \Delta y)^2}{4 \max(\alpha_{i,j})}
\end{equation}

In practice for the implementation it is advantageous to specify an
optional parameter $C$, $C \in [0, 1]$ so that the user can restrict
the "inner time stepping":

#+NAME: eqn:CFL2DFTCS_impl
\begin{equation}
\Delta t \leq C \cdot \frac{\min(\Delta x, \Delta y)^2}{4 \max(\alpha_{i,j})}
\end{equation}

** Boundary conditions

In analogy to the treatment of the 1D homogeneous FTCS scheme (cfr
section 1), we need to differentiate the domain boundaries ($i=0$ and
$i=n_x$; the same applies to $j$ of course) accounting for the
discrepancy in the discretization.

For the zero-th (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,
equation [[eqn:CS_het]] becomes:

#+NAME: eqn:2D_FTCS_left
\begin{equation}
\begin{aligned}
\frac{\partial}{\partial x} \left(\alpha(x)
\frac{\partial C }{\partial x}\right)(x_0) & \simeq \frac{1}{\Delta x}\left[\alpha_{i+1/2} \left( \frac{C_{i+1} -C_{i}}{\Delta x} \right) - \alpha_{i} \left( \frac{C_{i} - l }{\frac{\Delta x}{2}} \right) \right]\\
&\displaystyle =\frac{1}{\Delta x^2} \left[ \alpha_{i+1/2}C_{i+1} - (\alpha_{i+1/2}+ 2\alpha_i) C_{i} + 2 \alpha_{i}\cdot l\right]
\end{aligned}
\end{equation}

\noindent Similarly, for $i=n_x$,

#+NAME: eqn:2D_FTCS_right
\begin{equation}
\begin{aligned}
\frac{\partial}{\partial x} \left(\alpha(x)
\frac{\partial C }{\partial x}\right)(x_n) & \simeq \frac{1}{\Delta x}\left[\alpha_{i} \left( \frac{r -C_{i}}{\frac{\Delta x}{2}} \right) - \alpha_{i-1/2} \left( \frac{C_{i} - C_{i-1}} {\Delta x} \right) \right]\\
&\displaystyle =\frac{1}{\Delta x^2} \left[ 2 \alpha_{i} r - (\alpha_{i+1/2}+ 2\alpha_i) C_{i} + \alpha_{i-1/2}\cdot C_{i-1}\right]
\end{aligned}
\end{equation}

If on the right boundary we have *closed* or Neumann condition, the
left derivative becomes zero and we are left with:

#+NAME: eqn:2D_FTCS_rightclosed
\begin{equation}
\begin{aligned}
\frac{\partial}{\partial x} \left(\alpha(x)
\frac{\partial C }{\partial x}\right)(x_n) & \simeq \frac{1}{\Delta x}\left[\cancel{\alpha_{i+1/2} \left( \frac{C_{i+1} -C_{i}}{\Delta x} \right)} - \alpha_{i-1/2} \left( \frac{C_{i} -C_{i-1}}{\Delta x} \right) \right]\\
&\displaystyle =\frac{\alpha_{i-1/2}}{\Delta x^2} (C_{i-1} - C_i)
\end{aligned}
\end{equation}