s_algorithm/text/notation.tex

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\section{Notations} 

The notations we use to discribe the discrete formulation 
of the model are summarised hereafter:\\
general notation:
\\ $\Delta x, \Delta y, \Delta r$ grid spacing in X,Y,R directions.
\\ $A_o$ : Area of the face orthogonal to "o" direction (o=u,v,w ...).
\\ ${\cal V}_u , {\cal V}_v , {\cal V}_v , {\cal V}_\theta$ :
Volume of the grid box surrounding $u,v,w,\theta$ point;
\\ $i,j,k$ : current index relative to X,Y,R directions;
\\basic operator:
\\ $\delta_i $ : $\delta_i \Phi = \Phi_{i+1} - \Phi_i $
\\ $\overline{~}i$ : $\overline{\Phi}^i = ( \Phi_{i+1} + \Phi_i ) / 2 $ 
\\ $\delta_x $ : $\delta_x \Phi = \frac{1}{\Delta x} \delta_i \Phi $
\\
\\ $\overline{\nabla}$ = gradient operator :  
$\overline{\nabla} \Phi = \{ \delta_x \Phi , \delta_y \Phi \}$
\\ $\overline{\nabla} \cdot$ = divergence operator :  
$\overline{\nabla}\cdot \vec{\mathrm{f}}  = 
\frac{1}{\cal V} \{ \delta_i A_x \mathrm{f}_x 
                  + \delta_j A_y \mathrm{f}_y \} $
\\ $\overline{\nabla}^2 $ = Laplacien operator :
$ \overline{\nabla}^2 \Phi = 
   \overline{\nabla}\cdot \overline{\nabla}\Phi $
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4			\section{Notations}
5
6			The notations we use to discribe the discrete formulation
7			of the model are summarised hereafter:\\
8			general notation:
9			\\ $\Delta x, \Delta y, \Delta r$ grid spacing in X,Y,R directions.
10			\\ $A_o$ : Area of the face orthogonal to "o" direction (o=u,v,w ...).
11			\\ ${\cal V}_u , {\cal V}_v , {\cal V}_v , {\cal V}_\theta$ :
12			Volume of the grid box surrounding $u,v,w,\theta$ point;
13			\\ $i,j,k$ : current index relative to X,Y,R directions;
14			\\basic operator:
15			\\ $\delta_i $ : $\delta_i \Phi = \Phi_{i+1} - \Phi_i $
16			\\ $\overline{~}i$ : $\overline{\Phi}^i = ( \Phi_{i+1} + \Phi_i ) / 2 $
17			\\ $\delta_x $ : $\delta_x \Phi = \frac{1}{\Delta x} \delta_i \Phi $
18			\\
19			\\ $\overline{\nabla}$ = gradient operator :
20			$\overline{\nabla} \Phi = \{ \delta_x \Phi , \delta_y \Phi \}$
21			\\ $\overline{\nabla} \cdot$ = divergence operator :
22			$\overline{\nabla}\cdot \vec{\mathrm{f}} =
23			\frac{1}{\cal V} \{ \delta_i A_x \mathrm{f}_x
24			+ \delta_j A_y \mathrm{f}_y \} $
25			\\ $\overline{\nabla}^2 $ = Laplacien operator :
26			$ \overline{\nabla}^2 \Phi =
27			\overline{\nabla}\cdot \overline{\nabla}\Phi $