--- manual/s_algorithm/text/spatial-discrete.tex 2001/09/25 20:13:42 1.6 +++ manual/s_algorithm/text/spatial-discrete.tex 2001/10/24 15:21:27 1.9 @@ -1,4 +1,4 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/spatial-discrete.tex,v 1.6 2001/09/25 20:13:42 adcroft Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/spatial-discrete.tex,v 1.9 2001/10/24 15:21:27 cnh Exp $ % $Name: $ \section{Spatial discretization of the dynamical equations} @@ -17,13 +17,13 @@ The finite volume method is used to discretize the equations in space. The expression ``finite volume'' actually has two meanings; one -is the method of cut or instecting boundaries (shaved or lopped cells -in our terminology) and the other is non-linear interpolation methods -that can deal with non-smooth solutions such as shocks (i.e. flux -limiters for advection). Both make use of the integral form of the -conservation laws to which the {\it weak solution} is a solution on -each finite volume of (sub-domain). The weak solution can be -constructed outof piece-wise constant elements or be +is the method of embedded or intersecting boundaries (shaved or lopped +cells in our terminology) and the other is non-linear interpolation +methods that can deal with non-smooth solutions such as shocks +(i.e. flux limiters for advection). Both make use of the integral form +of the conservation laws to which the {\it weak solution} is a +solution on each finite volume of (sub-domain). The weak solution can +be constructed out of piece-wise constant elements or be differentiable. The differentiable equations can not be satisfied by piece-wise constant functions. @@ -67,7 +67,9 @@ \subsection{C grid staggering of variables} \begin{figure} -\centerline{ \resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}} } +\begin{center} +\resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}} +\end{center} \caption{Three dimensional staggering of velocity components. This facilitates the natural discretization of the continuity and tracer equations. } @@ -113,15 +115,18 @@ \subsection{Horizontal grid} +\label{sec:spatial_discrete_horizontal_grid} \begin{figure} -\centerline{ \begin{tabular}{cc} +\begin{center} +\begin{tabular}{cc} \raisebox{1.5in}{a)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Ac.eps}} & \raisebox{1.5in}{b)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Az.eps}} \\ \raisebox{1.5in}{c)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Au.eps}} & \raisebox{1.5in}{d)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Av.eps}} -\end{tabular} } +\end{tabular} +\end{center} \caption{ Staggering of horizontal grid descriptors (lengths and areas). The grid lines indicate the tracer cell boundaries and are the reference @@ -313,11 +318,13 @@ \subsection{Vertical grid} \begin{figure} -\centerline{ \begin{tabular}{cc} +\begin{center} + \begin{tabular}{cc} \raisebox{4in}{a)} \resizebox{!}{4in}{ \includegraphics{part2/vgrid-cellcentered.eps}} & \raisebox{4in}{b)} \resizebox{!}{4in}{ \includegraphics{part2/vgrid-accurate.eps}} -\end{tabular} } +\end{tabular} +\end{center} \caption{Two versions of the vertical grid. a) The cell centered approach where the interface depths are specified and the tracer points centered in between the interfaces. b) The interface centered @@ -382,9 +389,9 @@ \subsection{Topography: partially filled cells} \begin{figure} -\centerline{ +\begin{center} \resizebox{4.5in}{!}{\includegraphics{part2/vgrid-xz.eps}} -} +\end{center} \caption{ A schematic of the x-r plane showing the location of the non-dimensional fractions $h_c$ and $h_w$. The physical thickness of a