# LaTeX2HTML 2002-2-1 (1.70)
# Associate images original text with physical files.


$key = q/displaystyleleft.DP{psi}{t}right|_{Dvect{i}}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$\displaystyle \left.\DP{\psi }{t}\right\vert _{\Dvect{i}}^{n}$">|; 

$key = q/displaystyleleft.DP{psi}{t}right|_{i,j}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$\displaystyle \left.\DP{\psi }{t}\right\vert _{i,j}^{n}$">|; 

$key = q/displaystylepsi_{i}^{n}+left.DP{psi}{t}right|^{n}Deltat+left.frac{1}{2}DP[2]{psi}{r}right|^{n}(Deltat)^{2}+O(Deltat^{3});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$\displaystyle \psi _{i}^{n} + \left.\DP{\psi }{t}\right\vert^{n}
\Delta t
+ \left.\frac{1}{2}\DP[2]{\psi }{r}\right\vert^{n}(\Delta t)^{2}
+ O(\Delta t^{3})$">|; 

$key = q/{displaymath}psi_{i}^{n+1}=psi_{i}^{n}+left.DP{psi}{t}right|_{i}^{n}Deltat+frac{{psi}{t}right|_{i}^{n}(Deltat)^{2}+cdotcdotcdot.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="\begin{displaymath}
\psi _{i}^{n+1}=\psi _{i}^{n} + \left.\DP{\psi }{t}\right\v...
...}\right\vert _{i}^{n}
(\Delta t)^{2}
+ \cdot \cdot \cdot .
\end{displaymath}">|; 

$key = q/FH_{x},FH_{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$FH_{x}, FH_{y}$">|; 

$key = q/AtimesB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$A\times B$">|; 

$key = q/A,B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$A, B$">|; 

$key = q/displaystyleu_{i+frac{1}{2},j}^{n}(psi_{i,j}^{n}+psi_{i+1,j}^{n})frac{Deltat}{Deltax},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="$\displaystyle u_{i+\frac{1}{2},j}^{n}(\psi _{i,j}^{n} + \psi _{i+1,j}^{n})
\frac{\Delta t}{\Delta x},$">|; 

$key = q/displaystylefrac{1}{psi_{i,j+frac{1}{2}}^{*}}left[frac{1}{2}(|v_{i,j+frac{1}{2}}}}^{n})^{2})frac{psi_{i,j+1}^{*}-psi_{i,j}^{*}}{Deltay}right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$\displaystyle \frac{1}{\psi _{i,j+\frac{1}{2}}^{*}}
\left[\frac{1}{2}(\vert v_{...
...rac{1}{2}}^{n})^{2})
\frac{\psi _{i,j+1}^{*}-\psi _{i,j}^{*}}{\Delta y}
\right.$">|; 

$key = q/{displaymath}DP[2]{psi}{t}=-DP{}{x}left[uleft(-DP{}{x}(upsi)right)right]=DP{}{x}x}(upsi)right]=DP{}{x}left[u^{2}DP{psi}{x}right]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="499" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="\begin{displaymath}
\DP[2]{\psi }{t}
= - \DP{}{x}\left[ u \left( -\DP{}{x}(u\...
...\psi ) \right]
= \DP{}{x}\left[ u^{2} \DP{\psi }{x} \right]
\end{displaymath}">|; 

$key = q/{eqnarraystar}DP[2]{psi}{t}&=&-DP{}{x}left[uleft(-DP{}{x}(upsi)-DP{}{y}(vpsi)rigt[vpsileft(DP{u}{y}+DP{v}{x}right)right]{eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="612" HEIGHT="163" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="\begin{eqnarray*}
\DP[2]{\psi }{t}
&amp;=&amp; - \DP{}{x}\left
[u \left( -\DP{}{x}(u...
... \DP{}{y}\left[ v\psi \left(\DP{u}{y} + \DP{v}{x}\right)\right]
\end{eqnarray*}">|; 

$key = q/{displaymath}DP{psi}{t}+DP{}{x}(upsi)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="\begin{displaymath}
\DP{\psi }{t} + \DP{}{x}(u\psi )= 0,
\end{displaymath}">|; 

$key = q/{displaymath}psi_{i}^{*}=psi_{i}^{n}-{F(psi_{i}^{n},psi_{i+1}^{n},u_{i+frac{1}{2si_{i-1}^{n},psi_{i}^{n},u_{i-frac{1}{2}}^{n})}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="415" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="\begin{displaymath}
\psi _{i}^{*} = \psi _{i}^{n} -
\{F(\psi _{i}^{n},\psi _{...
...
F(\psi _{i-1}^{n},\psi _{i}^{n},u_{i-\frac{1}{2}}^{n})\}.
\end{displaymath}">|; 

$key = q/0leqxleq1,0leqyleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$0\leq x \leq 1, 0\leq y \leq 1$">|; 

$key = q/tilde{u}_{1+frac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$\tilde{u}_{1+\frac{1}{2}}$">|; 

$key = q/{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})}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="\begin{displaymath}
\DP{\psi }{t} = -
\{F(\psi _{i}^{n},\psi _{i+1}^{n},u_{i+\f...
...-
F(\psi _{i-1}^{n},\psi _{i}^{n},u_{i-\frac{1}{2}}^{n})\}.
\end{displaymath}">|; 

$key = q/{displaymath}DP{psi}{t}+DP{}{x}(upsi)+DP{}{y}(vpsi)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="\begin{displaymath}
\DP{\psi }{t} + \DP{}{x}(u\psi ) + \DP{}{y}(v\psi ) = 0,
\end{displaymath}">|; 

$key = q/{displaymath}DP{psi}{t}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="\begin{displaymath}
\DP{\psi }{t} = 0,
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{4}left(psi_{i+1,j+1}^{*}+psi_{i,j+1}^{*}+psi_{i+1,j}^{*}+psi_{i,j}^{*}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$\displaystyle \frac{1}{4}
\left( \psi _{i+1,j+1}^{*} + \psi _{i,j+1}^{*} +
\psi _{i+1,j}^{*} + \psi _{i,j}^{*} \right),$">|; 

$key = q/u_{d};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$u_{d}$">|; 

$key = q/{displaymath}F(psi_{i}^{n},psi_{i+1}^{n},u_{i+frac{1}{2}}^{n})=[(u_{i+frac{1}{2}frac{1}{2}}^{n}|)psi_{i+1}^{n}]frac{1}{2Deltax},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="538" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="\begin{displaymath}
F(\psi _{i}^{n},\psi _{i+1}^{n},u_{i+\frac{1}{2}}^{n})
= [...
...+\frac{1}{2}}^{n}\vert)\psi _{i+1}^{n}]
\frac{1}{2\Delta x},
\end{displaymath}">|; 

$key = q/{displaymath}tilde{u}rightarrow0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="\begin{displaymath}
\tilde{u} \rightarrow 0,
\end{displaymath}">|; 

$key = q/textstyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$\textstyle =$">|; 

$key = q/displaystyle+DP{}{x}left.left[frac{1}{2}(|u|Deltax-Deltatu^{2})DP{psi}{x}-frac{1}{2}DeltatuvDP{psi}{y}right]right|_{i,j}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="365" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$\displaystyle + \DP{}{x}\left.\left[\frac{1}{2}(\vert u\vert\Delta x-\Delta t u...
...P{\psi }{x} - \frac{1}{2}\Delta t uv\DP{\psi }{y}\right]
\right\vert _{i,j}^{n}$">|; 

$key = q/{eqnarraystar}F_{x}(psi_{i,j}^{n},psi_{i+1,j}^{n},u_{i+frac{1}{2},j}^{n})&=&[(u_|)psi_{i,j+1}^{n}]frac{Deltat}{2Deltay},{eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="646" HEIGHT="94" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="\begin{eqnarray*}
F_{x}(\psi _{i,j}^{n},\psi _{i+1,j}^{n},u_{i+\frac{1}{2},j}^{...
...}{2}}^{n}\vert)\psi _{i,j+1}^{n}]
\frac{\Delta t}{2\Delta y},
\end{eqnarray*}">|; 

$key = q/displaystyletilde{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$\displaystyle \tilde{v}$">|; 

$key = q/displaystylepsi_{i,j}^{*}-{F_{x}(psi_{i,j}^{*},psi_{i+1,j}^{*},tilde{u}_{i+frac{{x}(psi_{i-1,j}^{*},psi_{i,j}^{*},tilde{u}_{i-frac{1}{2},j})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$\displaystyle \psi _{i,j}^{*} -
\{F_{x}(\psi _{i,j}^{*},\psi _{i+1,j}^{*},\tild...
...2},j})
- F_{x}(\psi _{i-1,j}^{*},\psi _{i,j}^{*},\tilde{u}_{i-\frac{1}{2},j})\}$">|; 

$key = q/displaystylefrac{1}{psi}left[frac{1}{2}(|u|Deltax-Deltatu^{2})DP{psi}{x}-frac{1}{psi}{y}right]-frac{1}{2}Deltatuleft(DP{u}{x}+DP{v}{y}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="476" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$\displaystyle \frac{1}{\psi }
\left[\frac{1}{2}(\vert u\vert\Delta x-\Delta t u...
...\DP{\psi }{y}\right]
- \frac{1}{2}\Delta t u\left(\DP{u}{x} + \DP{v}{y}\right),$">|; 

$key = q/displaystyle-left.DP{}{x}(upsi)right|_{i,j}^{n}+DP{}{x}left.left(frac{1}{2}|u|DeltaxDP{psi}{x}right)right|_{i,j}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$\displaystyle -\left.\DP{}{x}(u\psi )\right\vert _{i,j}^{n}
+ \DP{}{x}\left.\left(\frac{1}{2}\vert u\vert\Delta x\DP{\psi }{x}\right)
\right\vert _{i,j}^{n}$">|; 

$key = q/psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$\psi $">|; 

$key = q/{displaymath}frac{psi_{i}^{n+1}-psi_{i}^{n}}{Deltat}=-left.DP{}{x}(upsi)right|_{taxDP{psi}{x}right)right|_{i}^{n}+O(Deltax^{2}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="483" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="\begin{displaymath}
\frac{\psi _{i}^{n+1}-\psi _{i}^{n}}{\Delta t}=
-\left.\DP...
...\DP{\psi }{x}\right)
\right\vert _{i}^{n} +O(\Delta x^{2}) .
\end{displaymath}">|; 

$key = q/{displaymath}mbox{max}left[underbrace{frac{|u_{i+frac{1}{2}}|Deltat}{Deltax}}_{A_{i}^{*}+psi_{i+1}^{*}+epsilon)}}_{C}right]leq1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="471" HEIGHT="103" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="\begin{displaymath}
\mbox{max}\left[\underbrace{\frac{\vert u_{i+\frac{1}{2}}\v...
...(\psi _{i}^{*}+\psi _{i+1}^{*}+\epsilon )}}_{C}\right]\leq 1,
\end{displaymath}">|; 

$key = q/{displaymath}mbox{max}left(frac{|tilde{u}_{i+frac{1}{2}}|Deltat}{Deltax}right)leq1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="\begin{displaymath}
\mbox{max}\left(\frac{\vert\tilde{u}_{i+\frac{1}{2}}\vert\Delta t}
{\Delta x}\right) \leq 1,
\end{displaymath}">|; 

$key = q/displaystylepsi_{i,j}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$\displaystyle \psi _{i,j}^{n+1}$">|; 

$key = q/|u|_{i+frac{1}{2}}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$\vert u\vert _{i+\frac{1}{2}}^{n}$">|; 

$key = q/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}})}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="404" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$\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}})\}.$">|; 

$key = q/displaystylefrac{1}{psi}left[frac{1}{2}(|v|Deltay-Deltatv^{2})DP{psi}{y}-frac{1}{psi}{x}right]-frac{1}{2}Deltatvleft(DP{u}{x}+DP{v}{y}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="473" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="$\displaystyle \frac{1}{\psi }
\left[\frac{1}{2}(\vert v\vert\Delta y-\Delta t v...
... \DP{\psi }{x}\right]
- \frac{1}{2}\Delta tv\left(\DP{u}{x} + \DP{v}{y}\right),$">|; 

$key = q/{figure}center{{Depsf[][80mm]{ps-figslashinit.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="363" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][80mm]{ps-fig/init.ps}
\end{center} \end{figure}">|; 

$key = q/{displaymath}tilde{u}_{i+frac{1}{2}}=frac{(|u_{i+frac{1}{2}}^{n}|Deltax-Deltatu_*})}{(psi_{i}^{*}+psi_{i+1}^{*}+epsilon)Deltax}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="\begin{displaymath}
\tilde{u}_{i+\frac{1}{2}} = \frac{
(\vert u_{i+\frac{1}{2}...
...}^{*})}
{(\psi _{i}^{*}+\psi _{i+1}^{*}+\epsilon )\Delta x}.
\end{displaymath}">|; 

$key = q/displaystylefrac{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),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$\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),$">|; 

$key = q/{displaymath}DP{}{x}left(K_{impl}DP{psi}{x}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="\begin{displaymath}
\DP{}{x}\left(K_{impl}\DP{\psi }{x}\right),
\end{displaymath}">|; 

$key = q/displaystyletilde{v}_{i,j+frac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$\displaystyle \tilde{v}_{i,j+\frac{1}{2}}$">|; 

$key = q/displaystylefrac{1}{psi}left[frac{1}{2}(|u|Deltax-Deltatu^{2})DP{psi}{x}-frac{1}{2}DeltatuvDP{psi}{y}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$\displaystyle \frac{1}{\psi }
\left[\frac{1}{2}(\vert u\vert\Delta x-\Delta t u^{2})
\DP{\psi }{x} - \frac{1}{2}\Delta t uv \DP{\psi }{y}\right],$">|; 

$key = q/(;);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$( \; )$">|; 

$key = q/{displaymath}DP[2]{psi}{t}=-DP{}{t}left[left(DP{}{x}(upsi)right)right]=-DP{}{x}lP{u}{t}right]+DP{}{x}left[u^{2}DP{psi}{x}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="636" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="\begin{displaymath}
\DP[2]{\psi }{t}
= -\DP{}{t}\left[ \left( \DP{}{x}(u\psi ...
...{u}{t}\right] +
\DP{}{x}\left[ u^{2} \DP{\psi }{x} \right],
\end{displaymath}">|; 

$key = q/psi_{i}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$\psi _{i}^{n}$">|; 

$key = q/tilde{u},tilde{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$\tilde{u}, \tilde{v}$">|; 

$key = q/t+Deltatslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="$t+\Delta t/2$">|; 

$key = q/Dvect{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$\Dvect{v}$">|; 

$key = q/{eqnarraystar}&&-frac{1}{2Deltax}left[left(-|u_{i}|left.DP{psi}{x}right|_{i}^{n}i}{x}right)right|_{i}^{n}+O(Deltax^{2}).{eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="660" HEIGHT="222" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="\begin{eqnarray*}
&amp;&amp; -\frac{1}{2\Delta x}\left[\left(
-\vert u_{i}\vert\left.\...
... x
\DP{\psi }{x}\right)\right\vert _{i}^{n} + O(\Delta x^{2}).
\end{eqnarray*}">|; 

$key = q/{eqnarraystar}&&-frac{1}{Deltax}left[left(u_{i}^{n}psi_{i}^{n}+frac{u_{i}^{n}}{2{}{x}(upsi)right|_{i}^{n}+O(Deltax^{2}),{eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="615" HEIGHT="333" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="\begin{eqnarray*}
&amp;&amp; -\frac{1}{\Delta x}\left[\left(u_{i}^{n}\psi _{i}^{n}
+ ...
...\left.\DP{}{x}(u\psi )\right\vert _{i}^{n}
+ O(\Delta x^{2}),
\end{eqnarray*}">|; 

$key = q/displaystyle-{[CF_{y}cdotFH_{y}+(1-CF_{y})cdotFL_{y}]_{i,j+frac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="$\displaystyle - \{[CF_{y}\cdot FH_{y}+(1-CF_{y})\cdot FL_{y}]_{i,j+\frac{1}{2}}$">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$t$">|; 

$key = q/{displaymath}DP{psi}{t}+DP{}{x}(upsi)=DP{}{x}left(K_{impl}DP{psi}{x}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="\begin{displaymath}
\DP{\psi }{t} + \DP{}{x}(u\psi ) =
\DP{}{x}\left(K_{impl}\DP{\psi }{x}\right),
\end{displaymath}">|; 

$key = q/displaystyleDP{psi}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$\displaystyle \DP{psi }{t}$">|; 

$key = q/(x_{m},y_{m})=(0.75,0.5);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$(x_{m}, y_{m})=(0.75, 0.5)$">|; 

$key = q/displaystylefrac{1}{psi_{i+frac{1}{2},j}^{*}}left[frac{1}{2}(|u_{i+frac{1}{2},j}j}^{n})^{2})frac{psi_{i+1,j}^{*}-psi_{i,j}^{*}}{Deltax}right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$\displaystyle \frac{1}{\psi _{i+\frac{1}{2},j}^{*}}
\left[\frac{1}{2}(\vert u_{...
...c{1}{2},j}^{n})^{2})
\frac{\psi _{i+1,j}^{*}-\psi _{i,j}^{*}}{\Delta x}
\right.$">|; 

$key = q/Deltat,Deltaxrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$\Delta t, \Delta x
\rightarrow 0$">|; 

$key = q/displaystyle-left.DP{}{x}(upsi)right|_{i,j}^{n}-left.DP{}{y}(vpsi)right|_{i,j}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$\displaystyle -\left.\DP{}{x}(u\psi )\right\vert _{i,j}^{n}
-\left.\DP{}{y}(v\psi )\right\vert _{i,j}^{n}$">|; 

$key = q/displaystyleF_{y}(psi_{i.j}^{n},psi_{i,j+1}^{n},v_{i,j+frac{1}{2}}^{n});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$\displaystyle F_{y}(\psi _{i.j}^{n},\psi _{i,j+1}^{n},v_{i,j+\frac{1}{2}}^{n})$">|; 

$key = q/{displaymath}DP{psi}{t}=-left.DP{}{x}(upsi)right|_{i}^{n}+DP{}{x}left.left(frac{taxDP{psi}{x}right)right|_{i}^{n}+O(Deltax^{2}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="\begin{displaymath}
\DP{\psi }{t} =
-\left.\DP{}{x}(u\psi )\right\vert _{i}^{n...
...\DP{\psi }{x}\right)
\right\vert _{i}^{n} +O(\Delta x^{2}) .
\end{displaymath}">|; 

$key = q/C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$C$">|; 

$key = q/displaystyleu_{i,j+frac{1}{2}}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$\displaystyle u_{i,j+\frac{1}{2}}^{n}$">|; 

$key = q/displaystyle-left.frac{1}{2}Deltatu_{i+frac{1}{2},j}^{n}v_{i+frac{1}{2},j}^{n}fr}{2}}^{*}-psi_{i+frac{1}{2},j-frac{1}{2}}^{*}}{Deltay}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$\displaystyle - \left.\frac{1}{2}\Delta t u_{i+\frac{1}{2},j}^{n}
v_{i+\frac{1}...
...j+\frac{1}{2}}^{*}-
\psi _{i+\frac{1}{2},j-\frac{1}{2}}^{*}}
{\Delta y}\right],$">|; 

$key = q/|u|_{i}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$\vert u\vert _{i}^{n}$">|; 

$key = q/displaystyletilde{u};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$\displaystyle \tilde{u}$">|; 

$key = q/{displaymath}-DP{}{x}(u_{d}psi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="\begin{displaymath}
- \DP{}{x}(u_{d}\psi ),
\end{displaymath}">|; 

$key = q/displaystyle-frac{1}{Deltax}left(u_{i+frac{1}{2}}^{n}frac{psi_{i}^{n}+psi_{i+1}^-u_{i-frac{1}{2}}^{n}frac{psi_{i}^{n}+psi_{i-1}^{n}}{2}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$\displaystyle -\frac{1}{\Delta x}\left(u_{i+\frac{1}{2}}^{n}\frac{
\psi _{i}^{n...
...^{n}}{2} - u_{i-\frac{1}{2}}^{n}\frac{
\psi _{i}^{n}+\psi _{i-1}^{n}}{2}\right)$">|; 

$key = q/{displaymath}psi_{i}^{n+1}=psi_{i}^{*}-{F(psi_{i}^{*},psi_{i+1}^{*},tilde{u}_{i+{i-1}^{*},psi_{i}^{*},tilde{u}_{i-frac{1}{2}})}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="\begin{displaymath}
\psi _{i}^{n+1} = \psi _{i}^{*} -
\{F(\psi _{i}^{*},\psi ...
...(\psi _{i-1}^{*},\psi _{i}^{*},\tilde{u}_{i-\frac{1}{2}})\}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{psi^{n+1}-psi^{n}}{Deltat}=-DP{}{x}(u^{n+frac{1}{2}}psi^{n}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="\begin{displaymath}
\frac{\psi ^{n+1}-\psi ^{n}}{\Delta t } =
-\DP{}{x}(u^{n+\frac{1}{2}}\psi ^{n}),
\end{displaymath}">|; 

$key = q/u_{i+frac{1}{2}}^{n},u_{i-frac{1}{2}}^{n},psi_{i+1}^{n},psi_{i-1}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$u_{i+\frac{1}{2}}^{n}, u_{i-\frac{1}{2}}^{n}, \psi _
{i+1}^{n}, \psi _{i-1}^{n}$">|; 

$key = q/{eqnarraystar}DP[2]{psi}{t}&=&-DP{}{x}left[uleft(-DP{}{x}(upsi)-DP{}{y}(vpsi)rig}left[v^{2}DP{psi}{y}+uvDP{psi}{x}right]{eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="612" HEIGHT="109" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="\begin{eqnarray*}
\DP[2]{\psi }{t}
&amp;=&amp; - \DP{}{x}\left
[u \left( -\DP{}{x}(u...
... + \DP{}{y}\left[ v^{2} \DP{\psi }{y} + uv \DP{\psi }{x}\right]
\end{eqnarray*}">|; 

$key = q/sim;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$\sim $">|; 

$key = q/omega=0.1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$\omega = 0.1$">|; 

$key = q/{displaymath}mbox{max}left(frac{|u_{i+frac{1}{2}}|Deltat}{Deltax}right)leq1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="\begin{displaymath}
\mbox{max}\left(\frac{\vert u_{i+\frac{1}{2}}\vert\Delta t}
{\Delta x}\right) \leq 1,
\end{displaymath}">|; 

$key = q/(x_{0},y_{0})=(0.5,0.5);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="$(x_{0}, y_{0})=(0.5, 0.5)$">|; 

$key = q/u_{i}^{n},psi_{i}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$u_{i}^{n}, \psi _{i}^{n}$">|; 

$key = q/I;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$I$">|; 

$key = q/displaystyle+DP{}{y}left.left[frac{1}{2}(|v|Deltay-Deltatv^{2})DP{psi}{y}-frac{1}{2}DeltatuvDP{psi}{x}right]right|_{i,j}^{n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$\displaystyle + \DP{}{y}\left.\left[\frac{1}{2}(\vert v\vert\Delta y-\Delta t v...
...{\psi }{y} - \frac{1}{2}\Delta t uv\DP{\psi }{x}\right]
\right\vert _{i,j}^{n},$">|; 

$key = q/displaystylepsi_{i+frac{1}{2},j}^{*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$\displaystyle \psi _{i+\frac{1}{2},j}^{*}$">|; 

$key = q/displaystylepsi_{i,j}^{n}-{F_{x}(psi_{i,j}^{n},psi_{i+1,j}^{n},u_{i+frac{1}{2},j-F_{x}(psi_{i-1,j}^{n},psi_{i,j}^{n},u_{i-frac{1}{2},j}^{n})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$\displaystyle \psi _{i,j}^{n} -
\{F_{x}(\psi _{i,j}^{n},\psi _{i+1,j}^{n},u_{i+...
...2},j}^{n}) -
F_{x}(\psi _{i-1,j}^{n},\psi _{i,j}^{n},u_{i-\frac{1}{2},j}^{n})\}$">|; 

$key = q/displaystyle-[CF_{x}cdotFH_{x}+(1-CF_{x})cdotFL_{x}]_{i-frac{1}{2},j}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$\displaystyle - [CF_{x}\cdot FH_{x}+(1-CF_{x})\cdot FL_{x}]_{i-\frac{1}{2},j}\}$">|; 

$key = q/displaystyle-DP{}{x}(u^{n}psi^{n})-frac{1}{2}DeltatDP{}{x}left(psileft.DP{u}{t}right)right|^{n}+O(Deltat^{2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="$\displaystyle -\DP{}{x}(u^{n}\psi ^{n})
-\frac{1}{2}\Delta t \DP{}{x}\left(\psi
\left.\DP{u}{t}\right)\right\vert^{n} + O(\Delta t^{2})$">|; 

$key = q/Dvect{i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$\Dvect{i}$">|; 

$key = q/displaystyle+sum_{I=1}^{N}DP{}{x^{I}}left.left[frac{1}{2}(|u^{I}|Deltax^{I}-Delt}{2}Deltatu^{I}u^{J}DP{psi}{x^{J}}right]right|_{Dvect{i}}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="496" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$\displaystyle + \sum _{I=1}^{N}\DP{}{x^{I}}\left.\left[\frac{1}{2}
(\vert u^{I}...
...c{1}{2}\Delta t u^{I}u^{J}
\DP{\psi }{x^{J}}\right]\right\vert _{\Dvect{i}}^{n}$">|; 

$key = q/displaystyle-DP{}{x}(u^{n+frac{1}{2}}psi^{n});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="$\displaystyle -\DP{}{x}(u^{n+\frac{1}{2}}\psi ^{n})$">|; 

$key = q/displaystyle-sum_{I=1}^{N}DP{}{x^{I}}left.left[left(frac{1}{2}Deltatu^{I}sum_{J=1}^{N}DP{u^{J}}{x^{J}}right)psiright]right|_{Dvect{i}}^{n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$\displaystyle - \sum _{I=1}^{N}\DP{}{x^{I}}\left. \left[\left(
\frac{1}{2}\Delt...
...um _{J=1}^{N}\DP{u^{J}}{x^{J}}\right)
\psi \right]\right\vert _{\Dvect{i}}^{n},$">|; 

$key = q/u^{n+frac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="$u^{n+\frac{1}{2}}$">|; 

$key = q/displaystyle-frac{1}{2}left.DP{}{x}left[Deltatupsileft(DP{u}{x}+DP{v}{y}right)riDeltatvpsileft(DP{u}{x}+DP{v}{y}right)right]right|^{n}_{i,j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="521" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="$\displaystyle - \frac{1}{2}\left.\DP{}{x}\left[\Delta t u\psi
\left(\DP{u}{x} +...
...[\Delta t v\psi
\left(\DP{u}{x} + \DP{v}{y}\right)\right]\right\vert^{n}_{i,j}.$">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$u$">|; 

$key = q/displaystyle+sum_{I=1}^{N}DP{}{x^{I}}left.left[frac{1}{2}(|u^{I}|Deltax^{I}-Deltac{1}{2}Deltatu^{I}u^{J}DP{psi}{y}right]right|_{Dvect{i}}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="489" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="$\displaystyle + \sum _{I=1}^{N}\DP{}{x^{I}}\left.\left[\frac{1}{2}
(\vert u^{I}...
...\frac{1}{2}\Delta t u^{I}u^{J}
\DP{\psi }{y}\right]\right\vert _{\Dvect{i}}^{n}$">|; 

$key = q/displaystyle+DP{}{y}left.left[frac{1}{2}(|v|Deltay-Deltatv^{2})DP{psi}{y}-frac{1}{2}DeltatuvDP{psi}{x}right]right|_{i,j}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$\displaystyle + \DP{}{y}\left.\left[\frac{1}{2}(\vert v\vert\Delta y-\Delta t v...
...DP{\psi }{y} - \frac{1}{2}\Delta tuv\DP{\psi }{x}\right]
\right\vert _{i,j}^{n}$">|; 

$key = q/{figure}center{{Depsf[][80mm]{ps-figslashcenter2-1.ps}{{{{Depsf[][80mm]{ps-figslashcenter2-2.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="726" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][80mm]{ps-fig/center2-1.ps}
\Depsf[][80mm]{ps-fig/center2-2.ps}
\end{center} \end{figure}">|; 

$key = q/displaystyle-sum_{I=1}^{N}left.DP{}{x^{I}}(u^{I}psi)right|_{Dvect{i}}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$\displaystyle - \sum _{I=1}^{N}\left.\DP{}{x^{I}}(u^{I}\psi )
\right\vert _{\Dvect{i}}^{n}$">|; 

$key = q/CF_{x}=CF_{y}=0.8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$CF_{x}= CF_{y}=0.8$">|; 

$key = q/displaystylepsi_{i,j}^{*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$\displaystyle \psi _{i,j}^{*}$">|; 

$key = q/displaystylepsi_{i}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="$\displaystyle \psi _{i}^{n+1}$">|; 

$key = q/Deltat=0.1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$\Delta t=0.1$">|; 

$key = q/{displaymath}tilde{u}=-u_{d},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="\begin{displaymath}
\tilde{u}= -u_{d},
\end{displaymath}">|; 

$key = q/{figure}center{{Depsf[][80mm]{ps-figslashupstream1.ps}{{{{Depsf[][80mm]{ps-figslashupstream2.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="726" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][80mm]{ps-fig/upstream1.ps}
\Depsf[][80mm]{ps-fig/upstream2.ps}
\end{center} \end{figure}">|; 

$key = q/{displaymath}left.DP{psi}{t}right|_{i}^{n}=-left.DP{}{x}(upsi)right|_{i}^{n}+DP{x}right]right|_{i}^{n}+O(Deltax^{2},Deltat^{2}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="547" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="\begin{displaymath}
\left.\DP{\psi }{t}\right\vert _{i}^{n}
= -\left.\DP{}{x}(...
...{x}\right]\right\vert _{i}^{n} +O(\Delta x^{2},\Delta t^{2}).
\end{displaymath}">|; 

$key = q/{displaymath}u_{d}=left{array{{ccl}-frac{K_{impl}}{psi}DP{psi}{x},&mbox{if}&psi>00,&mbox{if}&psi=0.array{right.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="\begin{displaymath}
u_{d}=\left\{
\begin{array}{ccl}
-\frac{K_{impl}}{\psi}\D...
...&amp;\psi &gt;0 \\\\
0, &amp; \mbox{if} &amp; \psi =0.
\end{array} \right.
\end{displaymath}">|; 

$key = q/displaystyle-left.frac{1}{2}Deltatu_{i,j+frac{1}{2}}^{n}v_{i,j+frac{1}{2}}^{n}fr}{2}}^{*}-psi_{i-frac{1}{2},j+frac{1}{2}}^{*}}{Deltax}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$\displaystyle - \left.\frac{1}{2}\Delta t u_{i,j+\frac{1}{2}}^{n}
v_{i,j+\frac{...
...j+\frac{1}{2}}^{*}
-\psi _{i-\frac{1}{2},j+\frac{1}{2}}^{*}}
{\Delta x}\right].$">|; 

$key = q/epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$\epsilon $">|; 

$key = q/FL_{x},FL_{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="$FL_{x}, FL_{y}$">|; 

$key = q/x^{I},u^{I};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$x^{I},u^{I}$">|; 

$key = q/displaystyle-frac{1}{2Deltax}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];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$\displaystyle -\frac{1}{2\Delta x}\left[\vert u_{i+\frac{1}{2}}^{n}\vert
(\psi ...
...n}) - \vert u_{i-\frac{1}{2}}^{n}\vert
(\psi _{i-1}^{n} - \psi _{i}^{n})\right]$">|; 

$key = q/{displaymath}u^{n+frac{1}{2}}=frac{u^{n}+u^{n+1}}{2}+O(Deltat^{2}){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="\begin{displaymath}
u^{n+\frac{1}{2}}=\frac{u^{n}+u^{n+1}}{2}+O(\Delta t^{2})
\end{displaymath}">|; 

$key = q/{displaymath}u_{d,Dvect{i}}^{I}=left.0.5[|u^{I}Deltax^{I}-Deltat(u^{I})^{2}]frac^{J}frac{1}{psi}DP{psi}{x^{J}}right|_{Dvect{i}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="504" HEIGHT="65" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="\begin{displaymath}
u_{d,\Dvect{i}}^{I} = \left.
0.5[\vert u^{I}\Delta x^{I}-\...
...J}\frac{1}{\psi }
\DP{\psi }{x^{J}}\right\vert _{\Dvect{i}},
\end{displaymath}">|; 

$key = q/displaystylev_{i,j+frac{1}{2}}^{n}(psi_{i,j}^{n}+psi_{i,j+1}^{n})frac{Deltat}{Deltay},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$\displaystyle v_{i,j+\frac{1}{2}}^{n}(\psi _{i,j}^{n} + \psi _{i,j+1}^{n})
\frac{\Delta t}{\Delta y},$">|; 

$key = q/(uDeltatslashDeltax);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$(u\Delta t/\Delta x)$">|; 

$key = q/displaystylefrac{1}{8}left(psi_{i+1,j+1}^{*}+psi_{i,j+1}^{*}+2psi_{i+1,j}^{*}+2psi_{i,j}^{*}+psi_{i+1,j-1}^{*}+psi_{i,j-1}^{*}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$\displaystyle \frac{1}{8}
\left( \psi _{i+1,j+1}^{*} + \psi _{i,j+1}^{*} +
2\ps...
...+1,j}^{*} + 2\psi _{i,j}^{*} +
\psi _{i+1,j-1}^{*} + \psi _{i,j-1}^{*} \right),$">|; 

$key = q/psi_{i}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\psi _{i}^{n+1}$">|; 

$key = q/x=iDeltax,t=nDeltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$x=i\Delta x,
t=n\Delta t$">|; 

$key = q/textstyleequiv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$\textstyle \equiv$">|; 

$key = q/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})}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="404" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$\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})\}.$">|; 

$key = q/psi^{n+1},u^{n+frac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="$\psi ^{n+1}, u^{n+\frac{1}{2}}$">|; 

$key = q/CF_{x},CF_{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="$CF_{x}, CF_{y}$">|; 

$key = q/{displaymath}u_{d,Dvect{i}}^{I}=left.0.5[|u^{I}Deltax^{I}-Deltat(u^{I})^{2}]fracI}sum_{J=1}^{N}DP{u^{J}}{x^{J}}right|_{Dvect{i}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="648" HEIGHT="65" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="\begin{displaymath}
u_{d,\Dvect{i}}^{I} = \left.
0.5[\vert u^{I}\Delta x^{I}-\...
...{I}\sum _{J=1}^{N}\DP{u^{J}}{x^{J}}
\right\vert _{\Dvect{i}}
\end{displaymath}">|; 

$key = q/displaystyletilde{u}_{i+frac{1}{2},j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$\displaystyle \tilde{u}_{i+\frac{1}{2},j}$">|; 

$key = q/Deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$\Delta t$">|; 

$key = q/10^{-15};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$10^{-15}$">|; 

$key = q/tilde{u},tilde{u};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$\tilde{u}, \tilde{u}$">|; 

$key = q/{eqnarraystar}u_{i+frac{1}{2}}^{n}&=&u_{i}^{n}+left.DP{u}{x}right|_{i}^{n}frac{Dright|_{i}^{n}(Deltax)^{2}+cdotcdotcdot.{eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="332" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="\begin{eqnarray*}
u_{i+\frac{1}{2}}^{n}&amp;=&amp;u_{i}^{n}+\left.\DP{u}{x}\right\vert ...
...{x}\right\vert _{i}^{n}
(\Delta x)^{2}
+ \cdot \cdot \cdot .
\end{eqnarray*}">|; 

$key = q/displaystylepsi_{i}^{n}+left.DP{psi}{t}right|^{n}Deltat-frac{1}{2}(Deltat)^{2}DP{}{x}left(psileft.DP{u}{t}right)right|^{n}+O(Deltat^{3}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="401" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="$\displaystyle \psi _{i}^{n} + \left.\DP{\psi }{t}\right\vert^{n}\Delta t
- \fra...
...^{2}\DP{}{x}
\left(\psi \left.\DP{u}{t}\right)\right\vert^{n}+ O(\Delta t^{3}),$">|; 

$key = q/displaystyle-left.DP{}{y}(vpsi)right|_{i,j}^{n}+DP{}{y}left.left(frac{1}{2}|v|De{i,j}^{n}-frac{1}{2}left.DP[2]{psi}{t}right|_{i,j}^{n}Deltat.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="419" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$\displaystyle -\left.\DP{}{y}(v\psi )\right\vert _{i,j}^{n}
+ \DP{}{y}\left.\le...
..._{i,j}^{n} - \frac{1}{2}\left.\DP[2]{\psi }{t}\right\vert _{i,j}^{n}
\Delta t .$">|; 

$key = q/{displaymath}DP{psi}{t}+Ddiv(Dvect{v}psi)=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="\begin{displaymath}
\DP{\psi }{t} + \Ddiv (\Dvect{v}\psi ) = 0.
\end{displaymath}">|; 

$key = q/{figure}center{{Depsf[][80mm]{ps-figslashfct3.ps}{{{{Depsf[][80mm]{ps-figslashfct4.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="726" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][80mm]{ps-fig/fct3.ps}
\Depsf[][80mm]{ps-fig/fct4.ps}
\end{center} \end{figure}">|; 

$key = q/displaystylepsi_{i+frac{1}{2},j+frac{1}{2}}^{*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$\displaystyle \psi _{i+\frac{1}{2},j+\frac{1}{2}}^{*}$">|; 

$key = q/displaystyle-[CF_{y}cdotFH_{y}+(1-CF_{y})cdotFL_{y}]_{i,j-frac{1}{2}}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="$\displaystyle - [CF_{y}\cdot FH_{y}+(1-CF_{y})\cdot FL_{y}]_{i,j-\frac{1}{2}}\}$">|; 

$key = q/displaystyleF_{x}(psi_{i,j}^{n},psi_{i+1,j}^{n},u_{i+frac{1}{2},j}^{n});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="$\displaystyle F_{x}(\psi _{i,j}^{n},\psi _{i+1,j}^{n},u_{i+\frac{1}{2},j}^{n})$">|; 

$key = q/(K_{impl}=0.5[|u|Deltax-Deltatu^{2}]);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$(K_{impl} = 0.5[\vert u\vert\Delta x-\Delta t u^{2}])$">|; 

$key = q/[;];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$[ \; ]$">|; 

$key = q/displaystylefrac{1}{psi}left[frac{1}{2}(|v|Deltay-Deltatv^{2})DP{psi}{y}-frac{1}{2}DeltatuvDP{psi}{x}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$\displaystyle \frac{1}{\psi }
\left[\frac{1}{2}(\vert v\vert\Delta y-\Delta t v^{2})
\DP{\psi }{y} - \frac{1}{2}\Delta t uv \DP{\psi }{x}\right],$">|; 

$key = q/{figure}center{{Depsf[][80mm]{ps-figslashfct1.ps}{{{{Depsf[][80mm]{ps-figslashfct2.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="726" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][80mm]{ps-fig/fct1.ps}
\Depsf[][80mm]{ps-fig/fct2.ps}
\end{center} \end{figure}">|; 

$key = q/Deltat=0.05;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$\Delta t=0.05 $">|; 

$key = q/F_{x},F_{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$F_{x}, F_{y}$">|; 

$key = q/Deltax=Deltay=0.01;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$\Delta x = \Delta y
= 0.01$">|; 

$key = q/{figure}center{{Depsf[][80mm]{ps-figslashmpdata1.ps}{{{{Depsf[][80mm]{ps-figslashmpdata2.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="726" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][80mm]{ps-fig/mpdata1.ps}
\Depsf[][80mm]{ps-fig/mpdata2.ps}
\end{center} \end{figure}">|; 

$key = q/tilde{u};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$\tilde{u}$">|; 

$key = q/(u,v)=(-omega(y-y_{0}),omega(x-x_{0}));MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$(u, v) = ( - \omega (y-y_{0}),
\omega (x-x_{0}))$">|; 

$key = q/{displaymath}left.DP{psi}{t}right|^{n}=-DP{}{x}(u^{n}psi^{n})+left.DP{}{x}left(f{2}DP{psi^{n}}{x}right)right|^{n}+O(Deltat^{2}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="458" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="\begin{displaymath}
\left.\DP{\psi }{t}\right\vert^{n} =
-\DP{}{x}(u^{n}\psi^{...
...}
\DP{\psi ^{n}}{x}\right)\right\vert^{n}+ O(\Delta t^{2}),
\end{displaymath}">|; 

$key = q/displaystylepsi_{i,j}^{n}-{[CF_{x}cdotFH_{x}+(1-CF_{x})cdotFL_{x}]_{i+frac{1}{2},j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$\displaystyle \psi _{i,j}^{n} -
\{[CF_{x}\cdot FH_{x}+(1-CF_{x})\cdot FL_{x}]_{i+\frac{1}{2},j}$">|; 

$key = q/|u_{i+frac{1}{2}}^{n}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$\vert u_{i+\frac{1}{2}}^{n}\vert$">|; 

$key = q/{displaymath}DP{psi}{t}=-DP{}{x}(upsi)+DP{}{x}left(KDP{psi}{x}right),;K=frac{1}{2}|u|Deltax,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="\begin{displaymath}
\DP{\psi }{t} =
-\DP{}{x}(u\psi )
+ \DP{}{x}\left(K\DP{\psi }{x}\right)
, \; K =\frac{1}{2}\vert u\vert\Delta x ,
\end{displaymath}">|; 

1;