mirror of
https://git.gfz-potsdam.de/naaice/tug.git
synced 2025-12-13 17:38:23 +01:00
442 lines
18 KiB
Org Mode
442 lines
18 KiB
Org Mode
#+TITLE: Numerical solution of diffusion equation in 2D with ADI Scheme
|
|
#+LaTeX_CLASS_OPTIONS: [a4paper,10pt]
|
|
#+LATEX_HEADER: \usepackage{fullpage}
|
|
#+LATEX_HEADER: \usepackage{amsmath, systeme}
|
|
#+OPTIONS: toc:nil
|
|
|
|
|
|
* 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, [[eqn:1]] can
|
|
be rewritten:
|
|
|
|
#+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 ([[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 centered
|
|
approximation of $\frac{\partial}{\partial x} \left(\alpha(x)
|
|
\frac{\partial C }{\partial x}\right)$:
|
|
|
|
\begin{align*}
|
|
\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{align*}
|
|
|
|
\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} + \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_{i} + \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}
|