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\begin{document}
\pagestyle{empty}\thispagestyle{empty}\lthtmltypeout{}%
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\onecolumn

% !!! IMAGES START HERE !!!



\setlength{\parskip}{4mm}%

\setlength{\parskip}{4mm}
\stepcounter{section}
\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath27}%
\begin{displaymath}
       \DP{\psi }{t} + \Ddiv (\Dvect{v}\psi ) = 0.
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline826}%
$\psi $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline828}%
$\Dvect{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath35}%
\begin{displaymath}
       \DP{\psi }{t} + \DP{}{x}(u\psi )= 0,
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline830}%
$x=i\Delta x,
     t=n\Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline834}%
$\psi _{i}^{n}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath45}%
\begin{displaymath}
\DP{\psi }{t} = -
       \{F(\psi _{i}^{n},\psi _{i+1}^{n},u_{i+\frac{1}{2}}^{n}) - 
       F(\psi _{i-1}^{n},\psi _{i}^{n},u_{i-\frac{1}{2}}^{n})\}.
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath64}%
\begin{displaymath}
       F(\psi _{i}^{n},\psi _{i+1}^{n},u_{i+\frac{1}{2}}^{n})
       = [(u_{i+\frac{1}{2}}^{n}+|u_{i+\frac{1}{2}}^{n}|)\psi _{i}^{n} + 
          (u_{i+\frac{1}{2}}^{n}-|u_{i+\frac{1}{2}}^{n}|)\psi _{i+1}^{n}]
          \frac{1}{2\Delta x},
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3002}%
$\displaystyle \DP{psi }{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3003}%
$\textstyle =$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3004}%
$\displaystyle -\frac{1}{\Delta x}\left(u_{i+\frac{1}{2}}^{n}\frac{
\psi _{i}^{n}+\psi _{i+1}^{n}}{2} - u_{i-\frac{1}{2}}^{n}\frac{
\psi _{i}^{n}+\psi _{i-1}^{n}}{2}\right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3005}%
$\displaystyle -\frac{1}{2\Delta x}\left[|u_{i+\frac{1}{2}}^{n}|
(\psi _{i}^{n} - \psi _{i+1}^{n}) - |u_{i-\frac{1}{2}}^{n}|
(\psi _{i-1}^{n} - \psi _{i}^{n})\right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline836}%
$u_{i+\frac{1}{2}}^{n}, u_{i-\frac{1}{2}}^{n}, \psi _
     {i+1}^{n}, \psi _{i-1}^{n}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline838}%
$u_{i}^{n}, \psi _{i}^{n}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline840}%
$|u_{i+\frac{1}{2}}^{n}|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline842}%
$|u|_{i+\frac{1}{2}}^{n}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline844}%
$|u|_{i}^{n}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar156}%
\begin{eqnarray*}
       u_{i+\frac{1}{2}}^{n}&=&u_{i}^{n}+\left.\DP{u}{x}\right|_{i}^{n}
                  \frac{\Delta x}{2}
                  + \frac{1}{2!}\left.\DP[2]{u}{x}\right|_{i}^{n}
                  (\frac{\Delta x}{2})^{2}
                  + \cdot \cdot \cdot , \\
       u_{i-\frac{1}{2}}^{n}&=&u_{i}^{n}-\left.\DP{u}{x}\right|_{i}^{n}
                  \frac{\Delta x}{2}
                  + \frac{1}{2!}\left.\DP[2]{u}{x}\right|_{i}^{n}
                  (\frac{\Delta x}{2})^{2}
                  + \cdot \cdot \cdot , \\
       |u|_{i+\frac{1}{2}}^{n}&=&|u|_{i}^{n}+\left.\DP{|u|}{x}\right|_{i}^{n}
                  \frac{\Delta x}{2}
                  + \frac{1}{2!}\left.\DP[2]{|u|}{x}\right|_{i}^{n}
                  (\frac{\Delta x}{2})^{2}
                  + \cdot \cdot \cdot , \\
       |u|_{i-\frac{1}{2}}^{n}&=&|u|_{i}^{n}-\left.\DP{|u|}{x}\right|_{i}^{n}
                  \frac{\Delta x}{2}
                  + \frac{1}{2!}\left.\DP[2]{|u|}{x}\right|_{i}^{n}
                  (\frac{\Delta x}{2})^{2}
                  + \cdot \cdot \cdot , \\
       \psi _{i+1}^{n}&=&\psi _{i}^{n} + \left.\DP{\psi }{x}\right|_{i}^{n}
                  \Delta x
                  + \frac{1}{2!}\left.\DP[2]{\psi }{x}\right|_{i}^{n}
                  (\Delta x)^{2}
                  + \cdot \cdot \cdot , \\
       \psi _{i-1}^{n}&=&\psi _{i}^{n} - \left.\DP{\psi }{x}\right|_{i}^{n}
                  \Delta x
                  + \frac{1}{2!}\left.\DP[2]{\psi }{x}\right|_{i}^{n}
                  (\Delta x)^{2}
                  + \cdot \cdot \cdot . 
     \end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline846}%
$( \; )$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar269}%
\begin{eqnarray*}
       && -\frac{1}{\Delta x}\left[\left(u_{i}^{n}\psi _{i}^{n} 
         + \frac{u_{i}^{n}}{2}\left.\DP{\psi }{x}\right|_{i}^{n} \Delta x
         + \psi _{i}^{n}\left.\DP{u}{x}\right|_{i}^{n}\frac{\Delta x}{2}
          \right. \right. \\
       &&  + \left. \frac{u_{i}^{n}}{2}\left.\DP[2]{\psi }{x}\right|_{i}^{n}
          (\Delta x)^{2}
          + \left. \DP{u}{x}\right| _{i}^{n}
           \left. \DP{\psi }{x}\right| _{i}^{n}
         \frac{(\Delta x)^{2}}{2}
         + \frac{\psi _{n}^{i}}{2}\left.\DP[2]{u}{x}\right|_{i}^{n}
         \left(\frac{\Delta x}{2}\right)^{2}
         + O(\Delta x^{3}) \right) \\
       && -\left(u_{i}^{n}\psi _{i}^{n} 
         - \frac{1}{2}u_{i}^{n}\left.\DP{\psi }{x}\right|_{i}^{n} \Delta x
         - \psi _{i}^{n}\left.\DP{u}{x}\right|_{i}^{n}
         \frac{\Delta x}{2} \right. \\
       && +\left. \left. 
         \frac{u_{i}^{n}}{2}\left.\DP[2]{\psi }{x}\right|_{i}^{n}
          (\Delta x)^{2}
       + \left.\DP{u}{x}\right|_{i}^{n}\left.\DP{\psi }{x}\right|_{i}^{n}
         \frac{(\Delta x)^{2}}{2}
         + \frac{\psi _{n}^{i}}{2}\left.\DP[2]{u}{x}\right|_{i}^{n}
         \left(\frac{\Delta x}{2}\right)^{2}
         + O(\Delta x^{3}) \right) \right] \\
       &=& -\frac{1}{\Delta x}\left[
         u_{i}^{n}\left.\DP{\psi }{x}\right|_{i}^{n}\Delta x
         + \psi _{i}^{n}\left.\DP{u}{x}\right|_{i}^{n} \Delta x 
       + O(\Delta x^{3}) \right] \\
       &=& -\left.\DP{}{x}(u\psi )\right|_{i}^{n} 
       + O(\Delta x^{2}),
     \end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline848}%
$[ \; ]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar390}%
\begin{eqnarray*}
       && -\frac{1}{2\Delta x}\left[\left(
         -|u_{i}|\left.\DP{\psi }{x}\right|_{i}^{n}\Delta x
         -\frac{|u_{i}|}{2}\left.\DP[2]{\psi }{x}\right|_{i}^{n}(\Delta x)^{2}
         -\left.\DP{|u|}{x}\right|_{i}^{n}\left.\DP{\psi }{x}\right|_{i}^{n}
         \frac{(\Delta x)^{2}}{2} +O(\Delta x^{3}) \right)\right. \\
       && -\left.\left(
         -|u_{i}|\left.\DP{\psi }{x}\right|_{i}^{n}\Delta x
         +\frac{|u_{i}|}{2}\left.\DP[2]{\psi }{x}\right|_{i}^{n}(\Delta x)^{2}
         +\left.\DP{|u|}{x}\right|_{i}^{n}\left.\DP{\psi }{x}\right|_{i}^{n}
         \frac{(\Delta x)^{2}}{2} + O(\Delta x^{3}) \right)\right] \\
       &=& \frac{1}{2\Delta x}\left[
         |u_{i}|\left.\DP[2]{\psi }{x}\right|_{i}^{n}(\Delta x)^{2}+
         \left.\DP{|u|}{x}\right|_{i}^{n}\left.\DP{\psi }{x}\right|_{i}^{n}
         \Delta x^{2} + O(\Delta x^{3})\right] \\
       &=& \DP{}{x}\left.\left(\frac{1}{2}|u|\Delta x
         \DP{\psi }{x}\right)\right|_{i}^{n} + O(\Delta x^{2}).
     \end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath468}%
\begin{displaymath}
       \DP{\psi }{t} =
       -\left.\DP{}{x}(u\psi )\right|_{i}^{n}
         + \DP{}{x}\left.\left(\frac{1}{2}|u|\Delta x\DP{\psi }{x}\right)
         \right|_{i}^{n} +O(\Delta x^{2}) .
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath812}%
\begin{displaymath}
       \DP{\psi }{t} =
       -\DP{}{x}(u\psi )
         + \DP{}{x}\left(K\DP{\psi }{x}\right)
         , \; K =\frac{1}{2}|u|\Delta x ,
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath498}%
\begin{displaymath}
       \frac{\psi _{i}^{n+1}-\psi _{i}^{n}}{\Delta t}=
       -\left.\DP{}{x}(u\psi )\right|_{i}^{n}
         + \DP{}{x}\left.\left(\frac{1}{2}|u|\Delta x\DP{\psi }{x}\right)
         \right|_{i}^{n} +O(\Delta x^{2}) .
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline850}%
$\psi _{i}^{n+1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath813}%
\begin{displaymath}
       \psi _{i}^{n+1}=\psi _{i}^{n} + \left.\DP{\psi }{t}\right|_{i}^{n}
                  \Delta t
                  + \frac{1}{2!}\left.\DP[2]{\psi }{t}\right|_{i}^{n}
                  (\Delta t)^{2}
                  + \cdot \cdot \cdot . 
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline854}%
$t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath814}%
\begin{displaymath}
       \DP[2]{\psi }{t} 
       = - \DP{}{x}\left[  u \left( -\DP{}{x}(u\psi ) \right) \right]
       = \DP{}{x}\left[  u \DP{}{x}(u\psi ) \right] 
       = \DP{}{x}\left[  u^{2} \DP{\psi }{x} \right]
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline856}%
$u$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath556}%
\begin{displaymath}
       \left.\DP{\psi }{t}\right|_{i}^{n}
       = -\left.\DP{}{x}(u\psi )\right|_{i}^{n}
         + \DP{}{x}\left.\left[\frac{1}{2}(|u|\Delta x-\Delta t u^{2})
             \DP{\psi }{x}\right]\right|_{i}^{n} +O(\Delta x^{2},\Delta t^{2}).
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath815}%
\begin{displaymath}
       \DP{\psi }{t} + \DP{}{x}(u\psi ) =
         \DP{}{x}\left(K_{impl}\DP{\psi }{x}\right),  
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline858}%
$(K_{impl} = 0.5[|u|\Delta x-\Delta t u^{2}])$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath816}%
\begin{displaymath}
       \DP{}{x}\left(K_{impl}\DP{\psi }{x}\right),  
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath597}%
\begin{displaymath}
       - \DP{}{x}(u_{d}\psi ),
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline862}%
$u_{d}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath604}%
\begin{displaymath}
       u_{d}=\left\{
         \begin{array}{ccl}
           -\frac{K_{impl}}{\psi}\DP{\psi }{x}, & \mbox{if} &\psi >0 \\
           0, & \mbox{if} & \psi =0. 
         \end{array}
         \right.
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline866}%
$\tilde{u}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath619}%
\begin{displaymath}
       \tilde{u}= -u_{d},
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath627}%
\begin{displaymath}
       \psi _{i}^{*} = \psi _{i}^{n} - 
       \{F(\psi _{i}^{n},\psi _{i+1}^{n},u_{i+\frac{1}{2}}^{n}) - 
       F(\psi _{i-1}^{n},\psi _{i}^{n},u_{i-\frac{1}{2}}^{n})\}. 
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath649}%
\begin{displaymath}
       \psi _{i}^{n+1} = \psi _{i}^{*} - 
       \{F(\psi _{i}^{*},\psi _{i+1}^{*},\tilde{u}_{i+\frac{1}{2}}) - 
       F(\psi _{i-1}^{*},\psi _{i}^{*},\tilde{u}_{i-\frac{1}{2}})\}. 
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath670}%
\begin{displaymath}
       \tilde{u}_{i+\frac{1}{2}} = \frac{
       (|u_{i+\frac{1}{2}}^{n}|\Delta x - \Delta t u_{i+\frac{1}{2}}^{2})
       (\psi _{i+1}^{*}-\psi _{i}^{*})}
       {(\psi _{i}^{*}+\psi _{i+1}^{*}+\epsilon )\Delta x}.
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline870}%
$\epsilon $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline872}%
$10^{-15}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline874}%
$\Delta t, \Delta x
     \rightarrow 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath817}%
\begin{displaymath}
       \tilde{u} \rightarrow 0, 
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath818}%
\begin{displaymath}
       \DP{\psi }{t} = 0,
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath709}%
\begin{displaymath}
       \mbox{max}\left(\frac{|u_{i+\frac{1}{2}}|\Delta t}
         {\Delta x}\right) \leq 1,
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath717}%
\begin{displaymath}
       \mbox{max}\left(\frac{|\tilde{u}_{i+\frac{1}{2}}|\Delta t}
         {\Delta x}\right) \leq 1,
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline878}%
$\tilde{u}_{1+\frac{1}{2}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath729}%
\begin{displaymath}
       \mbox{max}\left[\underbrace{\frac{|u_{i+\frac{1}{2}}|\Delta t}
         {\Delta x}}_{A}\times \underbrace{
         \left( 1 - \frac{|u_{i+\frac{1}{2}}|\Delta t}{\Delta x}\right)}_{B}
         \times \underbrace{
         \frac{|\psi _{i+1}^{*}-\psi _{i}^{*}|}
       {(\psi _{i}^{*}+\psi _{i+1}^{*}+\epsilon )}}_{C}\right]\leq 1,
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline880}%
$A, B$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline882}%
$C$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline884}%
$A\times B$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3060}%
$\displaystyle \left.\DP{\psi }{t}\right|_{i,j}^{n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3062}%
$\displaystyle -\left.\DP{}{x}(u\psi )\right|_{i,j}^{n}
+ \DP{}{x}\left.\left(\frac{1}{2}|u|\Delta x\DP{\psi }{x}\right)
\right|_{i,j}^{n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3063}%
$\displaystyle -\left.\DP{}{y}(v\psi )\right|_{i,j}^{n}
+ \DP{}{y}\left.\left(\frac{1}{2}|v|\Delta y\DP{\psi }{y}\right)
\right|_{i,j}^{n} - \frac{1}{2}\left.\DP[2]{\psi }{t}\right|_{i,j}^{n}
\Delta t .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar1055}%
\begin{eqnarray*}
       \DP[2]{\psi }{t} 
       &=& - \DP{}{x}\left
         [u \left( -\DP{}{x}(u\psi ) -\DP{}{y}(v\psi ) \right) \right]
       - \DP{}{y}\left
         [v \left( -\DP{}{x}(u\psi ) -\DP{}{y}(v\psi ) \right) \right] \\
       &=& \DP{}{x}\left[  u^{2} \DP{\psi }{x} + uv \DP{\psi }{y}\right]
       + \DP{}{y}\left[  v^{2} \DP{\psi }{y} + uv \DP{\psi }{x}\right]
     \end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3069}%
$\displaystyle -\left.\DP{}{x}(u\psi )\right|_{i,j}^{n}
-\left.\DP{}{y}(v\psi )\right|_{i,j}^{n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3070}%
$\displaystyle + \DP{}{x}\left.\left[\frac{1}{2}(|u|\Delta x-\Delta t u^{2})
\DP{\psi }{x} - \frac{1}{2}\Delta t uv\DP{\psi }{y}\right]
\right|_{i,j}^{n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3071}%
$\displaystyle + \DP{}{y}\left.\left[\frac{1}{2}(|v|\Delta y-\Delta t v^{2})
\DP{\psi }{y} - \frac{1}{2}\Delta t uv\DP{\psi }{x}\right]
\right|_{i,j}^{n},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1421}%
$\tilde{u}, \tilde{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3074}%
$\displaystyle \tilde{u}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3075}%
$\textstyle \equiv$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3076}%
$\displaystyle \frac{1}{\psi }
\left[\frac{1}{2}(|u|\Delta x-\Delta t u^{2})
\DP{\psi }{x} - \frac{1}{2}\Delta t uv \DP{\psi }{y}\right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3077}%
$\displaystyle \tilde{v}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3079}%
$\displaystyle \frac{1}{\psi }
\left[\frac{1}{2}(|v|\Delta y-\Delta t v^{2})
\DP{\psi }{y} - \frac{1}{2}\Delta t uv \DP{\psi }{x}\right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3081}%
$\displaystyle \psi _{i,j}^{*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3083}%
$\displaystyle \psi _{i,j}^{n} -
\{F_{x}(\psi _{i,j}^{n},\psi _{i+1,j}^{n},u_{i+\frac{1}{2},j}^{n}) -
F_{x}(\psi _{i-1,j}^{n},\psi _{i,j}^{n},u_{i-\frac{1}{2},j}^{n})\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3084}%
$\displaystyle - \{F_{y}(\psi _{i,j}^{n},\psi _{i,j+1}^{n},v_{i,j+\frac{1}{2}}^{n})
- F_{y}(\psi _{i,j-1}^{n},\psi _{i,j}^{n},v_{i,j-\frac{1}{2}}^{n})\}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1423}%
$\tilde{u}, \tilde{u}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3087}%
$\displaystyle \psi _{i,j}^{n+1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3089}%
$\displaystyle \psi _{i,j}^{*} -
\{F_{x}(\psi _{i,j}^{*},\psi _{i+1,j}^{*},\tilde{u}_{i+\frac{1}{2},j})
- F_{x}(\psi _{i-1,j}^{*},\psi _{i,j}^{*},\tilde{u}_{i-\frac{1}{2},j})\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3090}%
$\displaystyle -\{F_{y}(\psi _{i,j}^{*},\psi _{i,j+1}^{*},
\tilde{v}_{i,j+\frac{1}{2}}) -
F_{y}(\psi _{i,j-1}^{*},\psi _{i,j}^{*},\tilde{v}_{i,j-\frac{1}{2}})\}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3092}%
$\displaystyle \tilde{u}_{i+\frac{1}{2},j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3094}%
$\displaystyle \frac{1}{\psi _{i+\frac{1}{2},j}^{*}}
\left[\frac{1}{2}(|u_{i+\frac{1}{2},j}^{n}|\Delta x-\Delta t
(u_{i+\frac{1}{2},j}^{n})^{2})
\frac{\psi _{i+1,j}^{*}-\psi _{i,j}^{*}}{\Delta x}
\right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3095}%
$\displaystyle - \left.\frac{1}{2}\Delta t u_{i+\frac{1}{2},j}^{n}
v_{i+\frac{1}{2},j}^{n}
\frac{\psi _{i+\frac{1}{2},j+\frac{1}{2}}^{*}-
\psi _{i+\frac{1}{2},j-\frac{1}{2}}^{*}}
{\Delta y}\right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3096}%
$\displaystyle \tilde{v}_{i,j+\frac{1}{2}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3098}%
$\displaystyle \frac{1}{\psi _{i,j+\frac{1}{2}}^{*}}
\left[\frac{1}{2}(|v_{i,j+\frac{1}{2}}^{n}|\Delta y-\Delta t
(v_{i,j+\frac{1}{2}}^{n})^{2})
\frac{\psi _{i,j+1}^{*}-\psi _{i,j}^{*}}{\Delta y}
\right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3099}%
$\displaystyle - \left.\frac{1}{2}\Delta t u_{i,j+\frac{1}{2}}^{n}
v_{i,j+\frac{1}{2}}^{n}
\frac{\psi _{i+\frac{1}{2},j+\frac{1}{2}}^{*}
-\psi _{i-\frac{1}{2},j+\frac{1}{2}}^{*}}
{\Delta x}\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3101}%
$\displaystyle \psi _{i+\frac{1}{2},j}^{*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3103}%
$\displaystyle \frac{1}{8}
\left( \psi _{i+1,j+1}^{*} + \psi _{i,j+1}^{*} +
2\psi _{i+1,j}^{*} + 2\psi _{i,j}^{*} +
\psi _{i+1,j-1}^{*} + \psi _{i,j-1}^{*} \right),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3104}%
$\displaystyle \psi _{i+\frac{1}{2},j+\frac{1}{2}}^{*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3106}%
$\displaystyle \frac{1}{4}
\left( \psi _{i+1,j+1}^{*} + \psi _{i,j+1}^{*} +
\psi _{i+1,j}^{*} + \psi _{i,j}^{*} \right),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3107}%
$\displaystyle u_{i,j+\frac{1}{2}}^{n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3109}%
$\displaystyle \frac{1}{4}
\left( u_{i+\frac{1}{2},j+1}^{n} + u_{i-\frac{1}{2},j+1}^{n} +
u_{i+\frac{1}{2},j}^{n} + u_{i-\frac{1}{2},j}^{n} \right),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmldisplayA{displaymath1464}%
\begin{displaymath}
      \DP{\psi }{t} + \DP{}{x}(u\psi ) + \DP{}{y}(v\psi ) = 0,
    \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1796}%
$0\leq x \leq 1, 0\leq y \leq 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1798}%
$\Delta x = \Delta y
        = 0.01$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1800}%
$(x_{0}, y_{0})=(0.5, 0.5)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1802}%
$\omega = 0.1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1804}%
$(u, v) = ( - \omega (y-y_{0}), 
        \omega (x-x_{0}))$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1808}%
$(x_{m}, y_{m})=(0.75, 0.5)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1810}%
$\Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1812}%
$(u\Delta t/\Delta x)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure1483}%
\begin{figure}      \begin{center}
        \Depsf[][80mm]{ps-fig/init.ps}
              \end{center}
    \end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{eqnarraystar1531}%
\begin{eqnarray*}
      F_{x}(\psi _{i,j}^{n},\psi _{i+1,j}^{n},u_{i+\frac{1}{2},j}^{n})&=& 
      [(u_{i+\frac{1}{2},j}^{n}+|u_{i+\frac{1}{2},j}^{n}|)\psi _{i,j}^{n} + 
          (u_{i+\frac{1}{2},j}^{n}-|u_{i+\frac{1}{2},j}^{n}|)\psi _{i+1,j}^{n}]
          \frac{\Delta t}{2\Delta x}, \\
      F_{y}(\psi _{i.j}^{n},\psi _{i,j+1}^{n},v_{i,j+\frac{1}{2}}^{n})&=&
      [(v_{i,j+\frac{1}{2}}^{n}+|v_{i,j+\frac{1}{2}}^{n}|)\psi _{i,j}^{n} + 
          (v_{i,j+\frac{1}{2}}^{n}-|v_{i,j+\frac{1}{2}}^{n}|)\psi _{i,j+1}^{n}]
          \frac{\Delta t}{2\Delta y}, 
     \end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure1587}%
\begin{figure}       \begin{center}
         \Depsf[][80mm]{ps-fig/upstream1.ps}
                  \Depsf[][80mm]{ps-fig/upstream2.ps}
                \end{center}
     \end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1814}%
$F_{x}, F_{y}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3145}%
$\displaystyle F_{x}(\psi _{i,j}^{n},\psi _{i+1,j}^{n},u_{i+\frac{1}{2},j}^{n})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3147}%
$\displaystyle u_{i+\frac{1}{2},j}^{n}(\psi _{i,j}^{n} + \psi _{i+1,j}^{n})
\frac{\Delta t}{\Delta x},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3148}%
$\displaystyle F_{y}(\psi _{i.j}^{n},\psi _{i,j+1}^{n},v_{i,j+\frac{1}{2}}^{n})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3150}%
$\displaystyle v_{i,j+\frac{1}{2}}^{n}(\psi _{i,j}^{n} + \psi _{i,j+1}^{n})
\frac{\Delta t}{\Delta y},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1816}%
$\Delta t=0.1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1818}%
$\Delta t=0.05 $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure1679}%
\begin{figure}       \begin{center}
         \Depsf[][80mm]{ps-fig/center2-1.ps}
                  \Depsf[][80mm]{ps-fig/center2-2.ps}
                \end{center}
     \end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1822}%
$\sim $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure1696}%
\begin{figure}       \begin{center}
         \Depsf[][80mm]{ps-fig/mpdata1.ps}
                  \Depsf[][80mm]{ps-fig/mpdata2.ps}
                \end{center}
     \end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3172}%
$\displaystyle \psi _{i,j}^{n} -
\{[CF_{x}\cdot FH_{x}+(1-CF_{x})\cdot FL_{x}]_{i+\frac{1}{2},j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3173}%
$\displaystyle - [CF_{x}\cdot FH_{x}+(1-CF_{x})\cdot FL_{x}]_{i-\frac{1}{2},j}\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3174}%
$\displaystyle - \{[CF_{y}\cdot FH_{y}+(1-CF_{y})\cdot FL_{y}]_{i,j+\frac{1}{2}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3175}%
$\displaystyle - [CF_{y}\cdot FH_{y}+(1-CF_{y})\cdot FL_{y}]_{i,j-\frac{1}{2}}\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1828}%
$FH_{x}, FH_{y}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1830}%
$FL_{x}, FL_{y}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1832}%
$CF_{x}, CF_{y}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1834}%
$CF_{x}= CF_{y}=0.8$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure1750}%
\begin{figure}       \begin{center}
         \Depsf[][80mm]{ps-fig/fct1.ps}
                  \Depsf[][80mm]{ps-fig/fct2.ps}
                \end{center}
     \end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure1760}%
\begin{figure}      \begin{center}
        \Depsf[][80mm]{ps-fig/fct3.ps}
                \Depsf[][80mm]{ps-fig/fct4.ps}
              \end{center}
    \end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2428}%
$I$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2430}%
$x^{I},u^{I}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2432}%
$\Dvect{i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3197}%
$\displaystyle \left.\DP{\psi }{t}\right|_{\Dvect{i}}^{n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3199}%
$\displaystyle - \sum _{I=1}^{N}\left.\DP{}{x^{I}}(u^{I}\psi )
\right|_{\Dvect{i}}^{n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3200}%
$\displaystyle + \sum _{I=1}^{N}\DP{}{x^{I}}\left.\left[\frac{1}{2}
(|u^{I}|\Delta x^{I}-\Delta t(u^{I})^{2})
\DP{\psi }{x^{I}}
- \sum _{J\neq I}^{N}\frac{1}{2}\Delta t u^{I}u^{J}
\DP{\psi }{x^{J}}\right]\right|_{\Dvect{i}}^{n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2001}%
\begin{displaymath}
       u_{d,\Dvect{i}}^{I} = \left.
           0.5[|u^{I}\Delta x^{I}-\Delta t(u^{I})^{2}]\frac{1}{\psi }
           \DP{\psi }{x^{I}}\right|_{\Dvect{i}}
           - \left.\sum _{J\neq I}^{N}0.5\Delta tu^{I}u^{J}\frac{1}{\psi }
           \DP{\psi }{x^{J}}\right|_{\Dvect{i}},
     \end{displaymath}%
\lthtmldisplayZ
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\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{eqnarraystar2026}%
\begin{eqnarray*}
       \DP[2]{\psi }{t} 
       &=& - \DP{}{x}\left
         [u \left( -\DP{}{x}(u\psi ) -\DP{}{y}(v\psi ) \right) \right]
       - \DP{}{y}\left
         [v \left( -\DP{}{x}(u\psi ) -\DP{}{y}(v\psi ) \right) \right] \\
       &=& \DP{}{x}\left[  u^{2} \DP{\psi }{x} + uv \DP{\psi }{y}\right]
       + \DP{}{y}\left[  v^{2} \DP{\psi }{y} + uv \DP{\psi }{x}\right] \\
       && +\DP{}{x}\left[u\psi \left(\DP{u}{x} + \DP{v}{y}\right)\right]
       + \DP{}{y}\left[ v\psi \left(\DP{u}{y} + \DP{v}{x}\right)\right]
     \end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3209}%
$\displaystyle + \DP{}{y}\left.\left[\frac{1}{2}(|v|\Delta y-\Delta t v^{2})
\DP{\psi }{y} - \frac{1}{2}\Delta tuv\DP{\psi }{x}\right]
\right|_{i,j}^{n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3210}%
$\displaystyle - \frac{1}{2}\left.\DP{}{x}\left[\Delta t u\psi
\left(\DP{u}{x} + \DP{v}{y}\right)\right]\right|^{n}_{i,j}
- \frac{1}{2}\left.\DP{}{y}\left[\Delta t v\psi
\left(\DP{u}{x} + \DP{v}{y}\right)\right]\right|^{n}_{i,j}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3214}%
$\displaystyle \frac{1}{\psi }
\left[\frac{1}{2}(|u|\Delta x-\Delta t u^{2})
\DP{\psi }{x} - \frac{1}{2}uv \DP{\psi }{y}\right]
- \frac{1}{2}\Delta t u\left(\DP{u}{x} + \DP{v}{y}\right),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3217}%
$\displaystyle \frac{1}{\psi }
\left[\frac{1}{2}(|v|\Delta y-\Delta t v^{2})
\DP{\psi }{y} - \frac{1}{2}uv \DP{\psi }{x}\right]
- \frac{1}{2}\Delta tv\left(\DP{u}{x} + \DP{v}{y}\right),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3222}%
$\displaystyle + \sum _{I=1}^{N}\DP{}{x^{I}}\left.\left[\frac{1}{2}
(|u^{I}|\Delta x^{I}-\Delta t(u^{I})^{2})
\DP{\psi }{x^{I}}
- \sum _{J\neq I}^{N}\frac{1}{2}\Delta t u^{I}u^{J}
\DP{\psi }{y}\right]\right|_{\Dvect{i}}^{n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3223}%
$\displaystyle - \sum _{I=1}^{N}\DP{}{x^{I}}\left. \left[\left(
\frac{1}{2}\Delta t u^{I}\sum _{J=1}^{N}\DP{u^{J}}{x^{J}}\right)
\psi \right]\right|_{\Dvect{i}}^{n},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2219}%
\begin{displaymath}
       u_{d,\Dvect{i}}^{I} = \left.
           0.5[|u^{I}\Delta x^{I}-\Delta t(u^{I})^{2}]\frac{1}{\psi }
           \DP{\psi }{x^{I}}\right|_{\Dvect{i}}
           - \left.\sum _{J\neq I}^{N}0.5\Delta tu^{I}u^{J}\frac{1}{\psi }
           \DP{\psi }{x^{J}}\right|_{\Dvect{i}}
           - \left. 0.5 \Delta t u^{I}\sum _{J=1}^{N}\DP{u^{J}}{x^{J}}
             \right|_{\Dvect{i}}
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath2249}%
\begin{displaymath}
       \DP[2]{\psi }{t} 
       = -\DP{}{t}\left[  \left( \DP{}{x}(u\psi ) \right) \right]
       = -\DP{}{x}\left[ \psi \DP{u}{t}+u\DP{\psi }{t}\right]
       = -\DP{}{x}\left[ \psi \DP{u}{t}\right] + 
       \DP{}{x}\left[  u^{2} \DP{\psi }{x} \right],
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2274}%
\begin{displaymath}
       \frac{\psi ^{n+1}-\psi ^{n}}{\Delta t } = 
       -\DP{}{x}(u^{n+\frac{1}{2}}\psi ^{n}),
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2444}%
$\psi ^{n+1}, u^{n+\frac{1}{2}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3237}%
$\displaystyle \psi _{i}^{n+1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3239}%
$\displaystyle \psi _{i}^{n} + \left.\DP{\psi }{t}\right|^{n}
\Delta t
+ \left.\frac{1}{2}\DP[2]{\psi }{r}\right|^{n}(\Delta t)^{2}
+ O(\Delta t^{3})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3241}%
$\displaystyle \psi _{i}^{n} + \left.\DP{\psi }{t}\right|^{n}\Delta t
- \frac{1}{2}(\Delta t )^{2}\DP{}{x}
\left(\psi \left.\DP{u}{t}\right)\right|^{n}+ O(\Delta t^{3}),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3242}%
$\displaystyle -\DP{}{x}(u^{n+\frac{1}{2}}\psi ^{n})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3244}%
$\displaystyle -\DP{}{x}(u^{n}\psi ^{n})
-\frac{1}{2}\Delta t \DP{}{x}\left(\psi
\left.\DP{u}{t}\right)\right|^{n} + O(\Delta t^{2})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2337}%
\begin{displaymath}
       \left.\DP{\psi }{t}\right|^{n} =
       -\DP{}{x}(u^{n}\psi^{n})+\left.\DP{}{x}\left(
           \frac{1}{2}\Delta t (u^{n})^{2} 
         \DP{\psi ^{n}}{x}\right)\right|^{n}+ O(\Delta t^{2}),
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2452}%
$t+\Delta t/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2454}%
$u^{n+\frac{1}{2}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2426}%
\begin{displaymath}
       u^{n+\frac{1}{2}}=\frac{u^{n}+u^{n+1}}{2}+O(\Delta t^{2})
     \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}

\end{document}