s_algorithm/text/notation.tex

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\subsection{Notation} 

The notations we use to describe the discrete formulation 
of the model are summarized 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+1/2} - \Phi_{i-1/2} $
\label{eq:delta_i}
\\ $\overline{~}i$ : $\overline{\Phi}^i = ( \Phi_{i+1/2} + \Phi_{i-1/2} ) / 2 $ 
\label{eq:bar_i}
\\ $\delta_x $ : $\delta_x \Phi = \frac{1}{\Delta x} \delta_i \Phi $
\label{eq:delta_x}
\\
\\ $\overline{\nabla}$ = gradient operator :  
$\overline{\nabla} \Phi = \{ \delta_x \Phi , \delta_y \Phi \}$
\label{eq:d_grad}
\\ $\overline{\nabla} \cdot$ = divergence operator :  
$\overline{\nabla}\cdot \vec{\mathrm{f}}  = 
\frac{1}{\cal A} \{ \delta_i \Delta y \mathrm{f}_x 
                  + \delta_j \Delta x \mathrm{f}_y \} $
\label{eq:d_div}
\\ $\overline{\nabla}^2 $ = Laplacian operator :
$ \overline{\nabla}^2 \Phi = 
   \overline{\nabla}\cdot \overline{\nabla}\Phi $
\label{eq:d_lap}
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2	% $Name: $
3
4	\subsection{Notation}
5
6	The notations we use to describe the discrete formulation
7	of the model are summarized 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+1/2} - \Phi_{i-1/2} $
16	\label{eq:delta_i}
17	\\ $\overline{~}i$ : $\overline{\Phi}^i = ( \Phi_{i+1/2} + \Phi_{i-1/2} ) / 2 $
18	\label{eq:bar_i}
19	\\ $\delta_x $ : $\delta_x \Phi = \frac{1}{\Delta x} \delta_i \Phi $
20	\label{eq:delta_x}
21	\\
22	\\ $\overline{\nabla}$ = gradient operator :
23	$\overline{\nabla} \Phi = \{ \delta_x \Phi , \delta_y \Phi \}$
24	\label{eq:d_grad}
25	\\ $\overline{\nabla} \cdot$ = divergence operator :
26	$\overline{\nabla}\cdot \vec{\mathrm{f}} =
27	\frac{1}{\cal A} \{ \delta_i \Delta y \mathrm{f}_x
28	+ \delta_j \Delta x \mathrm{f}_y \} $
29	\label{eq:d_div}
30	\\ $\overline{\nabla}^2 $ = Laplacian operator :
31	$ \overline{\nabla}^2 \Phi =
32	\overline{\nabla}\cdot \overline{\nabla}\Phi $
33	\label{eq:d_lap}