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Improved consistency of notation in ADI_scheme and reworked section 2D ADI
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Marco De Lucia
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@ -1,11 +1,13 @@
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#+TITLE: Numerical solution of diffusion equation in 2D with ADI Scheme
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#+LaTeX_CLASS_OPTIONS: [a4paper,10pt]
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#+LATEX_HEADER: \usepackage{fullpage}
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#+LATEX_HEADER: \usepackage{amsmath}
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#+LATEX_HEADER: \usepackage{amsmath, systeme}
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#+OPTIONS: toc:nil
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* Finite differences with nodes as cells' centres
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* Diffusion in 1D
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** Finite differences with nodes as cells' centres
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The 1D diffusion equation is:
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@ -34,7 +36,7 @@ $\alpha$, we can write:
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#+NAME: eqn:2
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\begin{equation}\displaystyle
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\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}
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\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}
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\end{equation}
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In practice, we evaluate the first derivatives of $C$ w.r.t. $x$ on
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@ -53,16 +55,16 @@ calling $l$ the numerical value of a constant boundary condition:
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#+NAME: eqn:3
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\begin{equation}\displaystyle
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\frac{C_0^{j+1} -C_0^{j}}{\Delta t} = \alpha\frac{\frac{C^j_{1}-C^j_{0}}{\Delta x}-
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\frac{C^j_{0}-l}{\frac{\Delta x}{2}}}{\Delta x}
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\frac{C_0^{t+1} -C_0^{t}}{\Delta t} = \alpha\frac{\frac{C^t_{1}-C^t_{0}}{\Delta x}-
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\frac{C^t_{0}-l}{\frac{\Delta x}{2}}}{\Delta x}
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\end{equation}
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This expression, once developed, yields:
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#+NAME: eqn:4
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\begin{align}\displaystyle
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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 \\
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& = C_0^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( C^j_{1}- 3 C^j_{0} +2l \right)
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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 \\
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& = C_0^{t} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( C^t_{1}- 3 C^t_{0} +2l \right)
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\end{align}
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@ -71,15 +73,15 @@ $C_n$ - calling $r$ the right boundary value - is:
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#+NAME: eqn:5
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\begin{equation}\displaystyle
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\frac{C_n^{j+1} -C_n^j}{\Delta t} = \alpha\frac{\frac{r - C^j_{n}}{\frac{\Delta x}{2}}-
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\frac{C^j_{n}-C^j_{n-1}}{\Delta x}}{\Delta x}
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\frac{C_n^{t+1} -C_n^t}{\Delta t} = \alpha\frac{\frac{r - C^t_{n}}{\frac{\Delta x}{2}}-
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\frac{C^t_{n}-C^t_{n-1}}{\Delta x}}{\Delta x}
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\end{equation}
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Which, developed, gives
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#+NAME: eqn:6
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\begin{align}\displaystyle
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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 \\
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& = C_n^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( 2 r - 3 C^j_{n} + C^j_{n-1} \right)
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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 \\
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& = C_n^{t} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( 2 r - 3 C^t_{n} + C^t_{n-1} \right)
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\end{align}
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If on the right boundary we have closed or Neumann condition, the left derivative in eq. [[eqn:5]]
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@ -88,7 +90,7 @@ becomes zero and we are left with:
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#+NAME: eqn:7
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\begin{equation}\displaystyle
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C_n^{j+1} = C_n^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot (C^j_{n-1} - C^j_n)
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C_n^{t+1} = C_n^{t} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot (C^t_{n-1} - C^t_n)
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\end{equation}
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@ -100,14 +102,14 @@ A similar treatment can be applied to the BTCS implicit scheme.
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First, we define the Backward Time difference:
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\begin{equation}
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\frac{\partial C^{j+1} }{\partial t} = \frac{C^{j+1}_i - C^{j}_i}{\Delta t}
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\frac{\partial C^{t+1} }{\partial t} = \frac{C^{t+1}_i - C^{t}_i}{\Delta t}
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\end{equation}
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Second the spatial derivative approximation, evaluated at time level $j+1$:
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Second the spatial derivative approximation, evaluated at time level $t+1$:
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#+NAME: eqn:secondderivative
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\begin{equation}
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\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}
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\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}
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\end{equation}
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Taking the 1D diffusion equation from [[eqn:1]] and substituting each term by the
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@ -123,8 +125,8 @@ equations given above leads to the following equation:
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#+NAME: eqn:1DBTCS
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\begin{align}\displaystyle
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\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 \\
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& = \alpha\frac{C^{j+1}_{i-1} - 2C^{j+1}_{i} + C^{j+1}_{i+1}}{\Delta x^2}
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\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 \\
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& = \alpha\frac{C^{t+1}_{i-1} - 2C^{t+1}_{i} + C^{t+1}_{i+1}}{\Delta x^2}
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\end{align}
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This only applies to inlet cells with no ghost node as neighbor. For the left
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@ -132,15 +134,15 @@ cell with its center at $\frac{dx}{2}$ and the constant concentration on the
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left ghost node called $l$ the equation goes as followed:
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\begin{equation}\displaystyle
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\frac{C_0^{j+1} -C_0^{j}}{\Delta t} = \alpha\frac{\frac{C^{j+1}_{1}-C^{j+1}_{0}}{\Delta x}-
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\frac{C^{j+1}_{0}-l}{\frac{\Delta x}{2}}}{\Delta x}
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\frac{C_0^{t+1} -C_0^{t}}{\Delta t} = \alpha\frac{\frac{C^{t+1}_{1}-C^{t+1}_{0}}{\Delta x}-
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\frac{C^{t+1}_{0}-l}{\frac{\Delta x}{2}}}{\Delta x}
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\end{equation}
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This expression, once developed, yields:
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\begin{align}\displaystyle
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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 \\
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& = C_0^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( C^{j+1}_{1}- 3 C^{j+1}_{0} +2l \right)
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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 \\
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& = C_0^{t} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( C^{t+1}_{1}- 3 C^{t+1}_{0} +2l \right)
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\end{align}
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Now we define variable $s_x$ as followed:
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@ -152,96 +154,122 @@ Now we define variable $s_x$ as followed:
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Substituting with the new variable $s_x$ and reordering of terms leads to the equation applicable to our model:
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\begin{equation}\displaystyle
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-C^j_0 = (2s_x) \cdot l + (-1 - 3s_x) \cdot C^{j+1}_0 + s_x \cdot C^{j+1}_1
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-C^t_0 = (2s_x) \cdot l + (-1 - 3s_x) \cdot C^{t+1}_0 + s_x \cdot C^{t+1}_1
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\end{equation}
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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 $r$:
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\begin{equation}\displaystyle
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\frac{C_n^{j+1} -C_n^{j}}{\Delta t} = \alpha\frac{\frac{r-C^{j+1}_{n}}{\frac{\Delta x}{2}}-
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\frac{C^{j+1}_{n}-C^{j+1}_{n-1}}{\Delta x}}{\Delta x}
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\frac{C_n^{t+1} -C_n^{t}}{\Delta t} = \alpha\frac{\frac{r-C^{t+1}_{n}}{\frac{\Delta x}{2}}-
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\frac{C^{t+1}_{n}-C^{t+1}_{n-1}}{\Delta x}}{\Delta x}
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\end{equation}
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This expression, once developed, yields:
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\begin{align}\displaystyle
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C_n^{j+1} & = C_n^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( 2r - 2C^{j+1}_{n} - C^{j+1}_{n} + C^{j+1}_{n-1} \right) \nonumber \\
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& = C_0^{j} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( 2r - 3C^{j+1}_{n} + C^{j+1}_{n-1} \right)
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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 \\
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& = C_0^{t} + \frac{\alpha \cdot \Delta t}{\Delta x^2} \cdot \left( 2r - 3C^{t+1}_{n} + C^{t+1}_{n-1} \right)
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\end{align}
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Now rearrange terms and substituting with $s_x$ leads to:
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\begin{equation}\displaystyle
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-C^j_n = s_x \cdot C^{j+1}_{n-1} + (-1 - 3s_x) \cdot C^{j+1}_n + (2s_x) \cdot r
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-C^t_n = s_x \cdot C^{t+1}_{n-1} + (-1 - 3s_x) \cdot C^{t+1}_n + (2s_x) \cdot r
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\end{equation}
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*TODO*
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- Tridiagonal matrix filling
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** 2D ADI using BTCS scheme
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#+LATEX: \clearpage
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In the previous sections we described the usage of FTCS and BTCS on 1D grids. To
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make usage of the BTCS scheme we are following the alternating-direction
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implicit method. Therefore we make use of second difference operator defined in
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equation [[eqn:secondderivative]] in both $x$ and $y$ direction for half the time
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step $\Delta t$. We are denoting the numerator of equation [[eqn:1DBTCS]] as
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$\delta^2_x C^n_{ij}$ and $\delta^2_y C^n_{ij}$ respectively with $i$ the
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position in $x$, $j$ the position in $y$ direction and $T$ as the current time
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step:
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* Diffusion in 2D: the Alternating Direction Implicit scheme
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\begin{align}\displaystyle
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\delta^2_x C^T_{ij} &= C^{T}_{i-1,j} - 2C^{T}_{i,j} + C^{T}_{i+1,j} \nonumber \\
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\delta^2_y C^T_{ij} &= C^{T}_{i,j-1} - 2C^{T}_{i,j} + C^{T}_{i,j+1}
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\end{align}
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Assuming a constant $\alpha_x$ and $\alpha_y$ in each direction the equation can
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be defined:
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#+NAME: eqn:genADI
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\begin{align}\displaystyle
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\frac{C^{T+1/2}_{ij}-C^T_{ij}}{\frac{\Delta t}{2}} &= \alpha_x \frac{\left( \delta^2_x C^{T+1/2}_{ij} + \delta^2_y C^{T}_{ij}\right)}{\Delta x^2} \nonumber \\
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\frac{C^{T+1}_{ij}-C^{T+1/2}_{ij}}{\frac{\Delta t}{2}} &= \alpha_y \frac{\left( \delta^2_x C^{T+1/2}_{ij} + \delta^2_y C^{T+1}_{ij}\right)}{\Delta y^2}
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\end{align}
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Now we will define $s_x$ and $s_y$ respectively as followed:
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\begin{align}\displaystyle
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s_x &= \frac{\alpha_x \cdot \frac{\Delta t}{2}}{\Delta x^2} \nonumber \\
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s_y &= \frac{\alpha_y \cdot \frac{\Delta t}{2}}{\Delta y^2}
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\end{align}
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Equation [[eqn:genADI]], once developed in the $x$ direction, yields:
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\begin{align}\displaystyle
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\frac{C^{T+1/2}_{ij}-C^T_{ij}}{\frac{\Delta t}{2}} &= \alpha_x \frac{\left( \delta^2_x C^{T+1/2}_{ij} + \delta^2_y C^{T}_{ij}\right)}{\Delta x^2} \nonumber \\
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\frac{C^{T+1/2}_{ij}-C^T_{ij}}{\frac{\Delta t}{2}} &= \alpha_x \frac{\left( \delta^2_x C^{T+1/2}_{ij} \right)}{\Delta x^2} + \alpha_x \frac{\left(\delta^2_y C^{T}_{ij}\right)}{\Delta x^2} \nonumber \\
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C^{T+1/2}_{ij}-C^T_{ij} &= s_x \delta^2_x C^{T+1/2}_{ij} + s_x \delta^2_y C^{T}_{ij} \nonumber \\
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-C^T_{ij} - s_x \delta^2_y C^{T}_{ij} &= C^{T+1/2}_{ij} + s_x \delta^2_x C^{T+1/2}_{ij}
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\end{align}
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All values on the left side of the equation are known and we are solving for
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$C^{1/2}$. The calculation in $y$ direction for another $\frac{\Delta t}{2}$ is
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similar and can also be easily derived. Therefore we do not show it here.
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Substituting $\delta$ in the equation leads to following epxression:
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In 2D, the diffusion equation (in absence of source terms) and
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assuming homogeneous but anisotropic diffusion coefficient
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$\alpha=(\alpha_x,\alpha_y)$ becomes:
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#+NAME: eqn:2d
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\begin{equation}
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-C^T_{ij} - s_x \left(C^T_{i,j-1}-2C^T_{i,j}+C^T_{i,j+1} \right) = C^{T+1/2}_{ij} + s_x \left( C^{T+1/2}_{i-1,j} - 2C^{T+1/2}_{i,j} + C^{T+1/2}_{i+1,j} \right)
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\displaystyle \frac{\partial C}{\partial t} = \alpha_x \frac{\partial^2 C}{\partial x^2} + \alpha_y\frac{\partial^2 C}{\partial y^2}
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\end{equation}
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This scheme also only applies to inlet cells without a relation to boundaries.
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Fortunately we already derived both cases of outer left and right inlet cell
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respectively. Hence we are able to redefine each $\delta^2$ case in x and y
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direction, assuming $l_x$ and $l_y$ the be the left boundary value and $r_x$ and
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$r_y$ the right one for each direction $x$ and $y$. The equations are exemplary
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for timestep $T+1/2$:
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** 2D ADI using BTCS scheme
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The Alternating Direction Implicit method consists in splitting the
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integration of eq. [[eqn:2d]] in two half-steps, each of which represents
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implicitly the derivatives in one direction, and explicitly in the
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other. Therefore we make use of second derivative operator defined in
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equation [[eqn:secondderivative]] in both $x$ and $y$ direction for half
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the time step $\Delta t$.
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Denoting $i$ the grid cell index along $x$ direction, $j$ the index in
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$y$ direction and $t$ as the time level, the spatially centered second
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derivatives can be written as:
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\begin{align}\displaystyle
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\delta^2_d C^{T+1/2}_{0,j} &= 2l_x - 3C^{T+1/2}_{0,j} + C^{T+1/2}_{1,j} \nonumber \\
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\delta^2_d C^{T+1/2}_{n,j} &= 2r_x - 3C^{T+1/2}_{n,j} + C^{T+1/2}_{n-1,j} \nonumber \\
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\delta^2_d C^{T+1/2}_{i,0} &= 2l_y - 3C^{T+1/2}_{i,0} + C^{T+1/2}_{i,1} \nonumber \\
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\delta^2_d C^{T+1/2}_{i,n} &= 2r_y - 3C^{T+1/2}_{i,n} + C^{T+1/2}_{i,n-1}
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\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} \\
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\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}
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\end{align}
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The ADI scheme is formally defined by the equations:
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#+NAME: eqn:genADI
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\begin{equation}
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\systeme{
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\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},
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\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}
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}
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\end{equation}
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\noindent The first of equations [[eqn:genADI]], which writes implicitly
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the spatial derivatives in $x$ direction, after bringing the $\Delta t
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/ 2$ terms on the right hand side and substituting $s_x=\frac{\alpha_x
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\cdot \Delta t}{2 \Delta x^2}$ and $s_y=\frac{\alpha_y\cdot\Delta
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t}{2\Delta y^2}$ reads:
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\begin{equation}\displaystyle
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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})
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\end{equation}
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\noindent Separating the known terms (at time level $t$) on the left
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hand side and the implicit terms (at time level $t+1/2$) on the right
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hand side, we get:
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#+NAME: eqn:sweepX
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\begin{equation}\displaystyle
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-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})
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\end{equation}
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\noindent Equation [[eqn:sweepX]] can be solved with a BTCS scheme since
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it corresponds to a tridiagonal system of equations, and resolves the
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$C^{t+1/2}$ at each inner grid cell.
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The second of equations [[eqn:genADI]] can be treated the same way and
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yields:
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#+NAME: eqn:sweepY
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\begin{equation}\displaystyle
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-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 inlet cells without a relation to
|
||||
boundaries. Fortunately we already derived both cases of outer left
|
||||
and right inlet cell respectively. Hence we are able to redefine each
|
||||
$\delta^2$ case in x and y direction, assuming $l_x$ and $l_y$ the be
|
||||
the left boundary value and $r_x$ and $r_y$ the right one for each
|
||||
direction $x$ and $y$. The equations are exemplary for time level
|
||||
$t+1/2$:
|
||||
|
||||
\begin{equation}
|
||||
\systeme{
|
||||
\displaystyle \delta^2_d C^{t+1/2}_{0,j} = 2l_x - 3C^{t+1/2}_{0,j} + C^{t+1/2}_{1,j} ,
|
||||
\displaystyle \delta^2_d C^{t+1/2}_{n,j} = 2r_x - 3C^{t+1/2}_{n,j} + C^{t+1/2}_{n-1,j} ,
|
||||
\displaystyle \delta^2_d C^{t+1/2}_{i,0} = 2l_y - 3C^{t+1/2}_{i,0} + C^{t+1/2}_{i,1} ,
|
||||
\displaystyle \delta^2_d C^{t+1/2}_{i,n} = 2r_y - 3C^{t+1/2}_{i,n} + C^{t+1/2}_{i,n-1}
|
||||
}
|
||||
\end{equation}
|
||||
|
||||
#+LATEX: \clearpage
|
||||
|
||||
* Old stuff
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user