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


$key = q/displaystylerho_{1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$\displaystyle \rho _{1}$">|; 

$key = q/{displaymath}eta_{i+frac{1}{2}}=frac{1}{2}epsilon_{i+frac{1}{2}}(1-epsilon_{i+frac{1}{2}}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="\begin{displaymath}
\eta _{i+\frac{1}{2}}=\frac{1}{2}\epsilon _{i+\frac{1}{2}}
(1-\epsilon _{i+\frac{1}{2}}),
\end{displaymath}">|; 

$key = q/R_{i}^{+},R_{i}^{-};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="$R_{i}^{+}, R_{i}^{-}$">|; 

$key = q/displaystyleQ_{i}^{-};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="$\displaystyle Q_{i}^{-}$">|; 

$key = q/displaystyleleft{array{{lcl}<mbox_mark>{min}(1,Q_{i.j}^{+}slashP_{i,j}^{+})<col_<col_mark><mbox_mark>{if}<col_mark>P_{i,j}^{+}=0.array{right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="$\displaystyle \left\{
\begin{array}{lcl}
\mbox{min}(1, Q_{i.j}^{+}/P_{i,j}^{+})...
...mbox{if} &amp; P_{i,j}^{+}&gt;0,
\\\\
0 &amp; \mbox{if} &amp; P_{i,j}^{+}=0.
\end{array}\right.$">|; 

$key = q/displaystyleQ_{i,j}^{-};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="$\displaystyle Q_{i,j}^{-}$">|; 

$key = q/{displaymath}rho_{3}=frac{rho_{i-1}^{n}Deltax}{Deltax+Deltat(u_{i}^{n}-u_{i-1}^{n})},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="\begin{displaymath}
\rho _{3} = \frac{\rho _{i-1}^{n}\Delta x}{\Delta x + \Delta t
(u_{i}^{n} - u_{i-1}^{n})},
\end{displaymath}">|; 

$key = q/{displaymath}Delta_{i+frac{1}{2}}=rho_{i+1}^{n+1}-rho_{i}^{n+1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="\begin{displaymath}
\Delta _{i+\frac{1}{2}} = \rho _{i+1}^{n+1}-\rho _{i}^{n+1},
\end{displaymath}">|; 

$key = q/displaystyleQ_{i,j}^{+};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="$\displaystyle Q_{i,j}^{+}$">|; 

$key = q/Deltax=0.01;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$\Delta x = 0.01$">|; 

$key = q/{displaymath}rho_{i}^{n+1}=rho_{i}^{n}-frac{epsilon}{2}(rho_{i+1}^{n}-rho_{i-1}^ight)(rho_{i+1}^{n}-2rho_{i}^{n}+rho_{i-1}^{n}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="483" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="\begin{displaymath}
\rho _{i}^{n+1} = \rho _{i}^{n} - \frac{\epsilon }{2}
(\rh...
...{2}\right)
(\rho _{i+1}^{n}-2\rho _{i}^{n}+\rho _{i-1}^{n}),
\end{displaymath}">|; 

$key = q/displaystylerho_{i}^{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$\displaystyle \rho _{i}^{b}$">|; 

$key = q/displaystyleP_{i}^{+};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$\displaystyle P_{i}^{+}$">|; 

$key = q/displaystyleA_{i,j+frac{1}{2}}(rho_{i,j+2}^{td}-rho_{i,j+1}^{td})<0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="$\displaystyle A_{i,j+\frac{1}{2}}(\rho _
{i,j+2}^{td} - \rho _{i,j+1}^{td}) &lt; 0,$">|; 

$key = q/displaystyle-{F_{y}(rho_{i,j}^{n},rho_{i,j+1}^{n},v_{i,j+frac{1}{2}}^{n})-F_{y}(rho_{i,j-1}^{n},rho_{i,j}^{n},v_{i,j-frac{1}{2}}^{n})}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$\displaystyle - \{F_{y}(\rho _{i,j}^{n},\rho _{i,j+1}^{n},v_{i,j+\frac{1}{2}}^{n})
- F_{y}(\rho _{i,j-1}^{n},\rho _{i,j}^{n},v_{i,j-\frac{1}{2}}^{n})\}.$">|; 

$key = q/{displaymath}rho_{i}^{n+1}=rho_{i}^{n}-frac{1}{Deltax}left{[C_{i+frac{1}{2}}F_{i(1-C_{i-frac{1}{2}})F_{i-frac{1}{2}}^{L}]right},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="647" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="\begin{displaymath}
\rho _{i}^{n+1}=\rho _{i}^{n}
- \frac{1}{\Delta x}\left\{[...
...}}^{H}+
(1-C_{i-\frac{1}{2}})F_{i-\frac{1}{2}}^{L}]\right\},
\end{displaymath}">|; 

$key = q/{figure}center{{Depsf[80mm][]{ps-figslashfig1.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="670" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="\begin{figure}\begin{center}
\Depsf[80mm][]{ps-fig/fig1.ps}
\end{center} \end{figure}">|; 

$key = q/displaystylembox{min}(rho_{i,j}^{n},;rho_{i,j}^{td}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="$\displaystyle \mbox{min}(\rho _{i,j}^{n}, \; \rho _
{i,j}^{td}),$">|; 

$key = q/displaystylefrac{1}{2}Q_{-}^{2}(rho_{i-1}^{n}-rho_{i}^{n})+frac{1}{2}Q_{+}^{2}(rho_{i+1}^{n}-rho_{i}^{n})+(Q_{+}+Q_{-})rho_{i}^{n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="421" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$\displaystyle \frac{1}{2}Q_{-}^{2}(\rho _{i-1}^{n}-\rho _{i}^{n})
+ \frac{1}{2}Q_{+}^{2}(\rho _{i+1}^{n}-\rho _{i}^{n})
+ (Q_{+}+Q_{-})\rho _{i}^{n},$">|; 

$key = q/displaystylembox{min}(rho_{i-1}^{td},;rho_{i}^{td},;rho_{i+1}^{td}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$\displaystyle \mbox{min}(\rho _{i-1}^{td}, \; \rho _
{i}^{td}, \; \rho _{i+1}^{td}).$">|; 

$key = q/displaystyle(rho_{i}-rho_{i}^{min})Deltax,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="$\displaystyle (\rho _{i}-\rho _{i}^{min})\Delta x,$">|; 

$key = q/displaystyleA_{i,j+frac{1}{2}}(rho_{i,j+1}^{td}-rho_{i,j}^{td})<0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="$\displaystyle A_{i,j+\frac{1}{2}}(\rho _
{i,j+1}^{td} - \rho _{i,j}^{td}) &lt; 0,$">|; 

$key = q/f_{i+frac{1}{2}}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$f_{i+\frac{1}{2}}&gt;0$">|; 

$key = q/displaystyle-[C_{i-frac{1}{2},j}F_{i-frac{1}{2},j}^{H}+(1-C_{i-frac{1}{2},j})F_{i-frac{1}{2},j}^{L}],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="$\displaystyle - [C_{i-\frac{1}{2},j}F_{i-\frac{1}{2},j}^{H}+
(1-C_{i-\frac{1}{2},j})F_{i-\frac{1}{2},j}^{L}],$">|; 

$key = q/displaystyle+mbox{max}(0,A_{i,j+frac{1}{2}})-mbox{min}(0,A_{i,j-frac{1}{2}}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="$\displaystyle + \mbox{max}(0, A_{i,j+\frac{1}{2}}) -
\mbox{min}(0,A_{i,j-\frac{1}{2}}),$">|; 

$key = q/{displaymath}rho_{i}^{n+1}=rho_{i}^{n}+frac{1}{8}(rho_{i+1}^{n}-2rho_{i}^{n}+rho_{i-1}^{n}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="\begin{displaymath}
\rho _{i}^{n+1} = \rho _{i}^{n}
+ \frac{1}{8}(\rho _{i+1}^{n}-2\rho _{i}^{n}
+\rho _{i-1}^{n}),
\end{displaymath}">|; 

$key = q/F_{i+frac{1}{2}}^{L};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$F_{i+\frac{1}{2}}^{L}$">|; 

$key = q/displaystyleR_{i}^{-};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="$\displaystyle R_{i}^{-}$">|; 

$key = q/C_{i+frac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$C_{i+\frac{1}{2}}$">|; 

$key = q/displaystylerho_{i,j}^{n}-{F_{x}(rho_{i,j}^{n},rho_{i+1,j}^{n},u_{i+frac{1}{2},j-F_{x}(rho_{i-1,j}^{n},rho_{i,j}^{n},u_{i-frac{1}{2},j}^{n})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="$\displaystyle \rho _{i,j}^{n} -
\{F_{x}(\rho _{i,j}^{n},\rho _{i+1,j}^{n},u_{i+...
...2},j}^{n}) -
F_{x}(\rho _{i-1,j}^{n},\rho _{i,j}^{n},u_{i-\frac{1}{2},j}^{n})\}$">|; 

$key = q/u=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$u=0$">|; 

$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|img185.png"
 ALT="$0\leq x \leq 1, 0\leq y \leq 1$">|; 

$key = q/eta_{i+frac{1}{2}}|Delta_{i+frac{1}{2}}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$\eta _{i+\frac{1}{2}}\vert\Delta _
{i+\frac{1}{2}}\vert$">|; 

$key = q/{displaymath}rho_{i}^{n+1}=tilde{rho}_{i}-(f_{i+frac{1}{2}}^{c}-f_{i-frac{1}{2}}^{c}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="\begin{displaymath}
\rho _{i}^{n+1}=\tilde{\rho }_{i} - (f_{i+\frac{1}{2}}^{c}-
f_{i-\frac{1}{2}}^{c}).
\end{displaymath}">|; 

$key = q/displaystyleA_{i+frac{1}{2},j}(rho_{i,j}^{td}-rho_{i-1,j}^{td})<0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="$\displaystyle A_{i+\frac{1}{2},j}(\rho _
{i,j}^{td} - \rho _{i-1,j}^{td}) &lt; 0,$">|; 

$key = q/{displaymath}f^{c}_{i+frac{1}{2}}equivS_{i+frac{1}{2}}mbox{max}{0,mbox{min}(S_{i1}{2}}|,;S_{i+frac{1}{2}}Delta_{i+frac{3}{2}})},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="\begin{displaymath}
f^{c}_{i+\frac{1}{2}} \equiv S_{i+\frac{1}{2}} \mbox{max}
...
...{1}{2}}\vert, \; S_{i+\frac{1}{2}}\Delta _{i+\frac{3}{2}})\},
\end{displaymath}">|; 

$key = q/0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$0.5$">|; 

$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/F_{i+frac{1}{2}}^{H};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$F_{i+\frac{1}{2}}^{H}$">|; 

$key = q/displaystylembox{min}(rho_{i-1,j}^{b},;rho_{i,j}^{b},;rho_{i+1,j}^{b},;rho_{i,j-1}^{b},;rho_{i,j-1}^{b}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="$\displaystyle \mbox{min}(\rho _{i-1,j}^{b}, \;
\rho _ {i,j}^{b}, \; \rho _{i+1,j}^{b}, \; \rho _{i,j-1}^{b}, \;
\rho _{i,j-1}^{b}),$">|; 

$key = q/{displaymath}f_{i+frac{1}{2}}=frac{1}{8}(rho_{i+1}^{n+1}-rho_{i}^{n+1}),;Delta_{i+frac{1}{2}}=rho_{i+1}^{n+1}-rho_{i}^{n+1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="\begin{displaymath}
f_{i+\frac{1}{2}}=
\frac{1}{8}(\rho _{i+1}^{n+1}-\rho _{i}...
...Delta _{i+\frac{1}{2}} = \rho _{i+1}^{n+1}-\rho _
{i}^{n+1},
\end{displaymath}">|; 

$key = q/displaystyleA_{i+frac{1}{2},j}(rho_{i+1,j}^{td}-rho_{i,j}^{td})<0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$\displaystyle A_{i+\frac{1}{2},j}(\rho _
{i+1,j}^{td} - \rho _{i,j}^{td}) &lt; 0,$">|; 

$key = q/displaystylepmfrac{1}{8}(rho_{ipm1}^{n+1}-rho_{i}^{n+1}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$\displaystyle \pm \frac{1}{8}(\rho _{i\pm
1}^{n+1}-\rho _{i}^{n+1}),$">|; 

$key = q/{figure}center{{Depsf[120mm][]{ps-figslashfig3.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="537" HEIGHT="332" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="\begin{figure}\begin{center}
\Depsf[120mm][]{ps-fig/fig3.ps}
\end{center} \end{figure}">|; 

$key = q/displaystylembox{min}(rho_{i-1}^{b},;rho_{i}^{b},;rho_{i+1}^{b}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$\displaystyle \mbox{min}(\rho _{i-1}^{b}, \; \rho _
{i}^{b}, \; \rho _{i+1}^{b}),$">|; 

$key = q/displaystylerho_{i.j}^{n}-frac{1}{DeltaxDeltay}{[C_{i+frac{1}{2},j}F_{i+frac{1}{2},j}^{H}+(1-C_{i+frac{1}{2},j})F_{i+frac{1}{2},j}^{L}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="392" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$\displaystyle \rho _{i.j}^{n} - \frac{1}{\Delta x\Delta
y}\{[C_{i+\frac{1}{2},j}F_{i+\frac{1}{2},j}^{H} +
(1-C_{i+\frac{1}{2},j})F_{i+\frac{1}{2},j}^{L}]$">|; 

$key = q/u=1.0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$u=1.0$">|; 

$key = q/t_{0}(equivnDeltat);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$t_{0} (\equiv n\Delta t)$">|; 

$key = q/displaystylerho_{i}^{n+1}-(f_{i+frac{1}{2}}-f_{i-frac{1}{2}}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$\displaystyle \rho _{i}^{n+1} -
(f_{i+\frac{1}{2}} - f_{i-\frac{1}{2}}),$">|; 

$key = q/t_{0}+Deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$t_{0}+\Delta t$">|; 

$key = q/{displaymath}rho_{i}^{n+1}=rho_{i}^{n}-frac{1}{Deltax}[F_{i+frac{1}{2}}-F_{i-frac{1}{2}}],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="\begin{displaymath}
\rho _{i}^{n+1}=\rho _{i}^{n} - \frac{1}{\Delta x}[F_{i+\frac{1}{2}}
-F_{i-\frac{1}{2}}],
\end{displaymath}">|; 

$key = q/displaystyleA_{i+frac{1}{2}}(rho_{i+2}^{td}-rho_{i+1}^{td})<0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$\displaystyle A_{i+\frac{1}{2}}(\rho _
{i+2}^{td} - \rho _{i+1}^{td}) &lt; 0,$">|; 

$key = q/displaystyleA_{i+frac{1}{2},j}(rho_{i+2,j}^{td}-rho_{i+1,j}^{td})<0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="$\displaystyle A_{i+\frac{1}{2},j}(\rho _
{i+2,j}^{td} - \rho _{i+1,j}^{td}) &lt; 0,$">|; 

$key = q/displaystylembox{格子点{i{に入る全てのantidiffusiveフラックスの和};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$\displaystyle \mbox{格子点 $i$ に入る全ての antidiffusive フ
ラックスの和}$">|; 

$key = q/displaystylembox{格子点{i,j{に入る全てのantidiffusiveフラックスの和};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$\displaystyle \mbox{格子点 $i,j$ に入る全ての antidiffusive フ
ラックスの和}$">|; 

$key = q/u_{i}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$u_{i}^{n}$">|; 

$key = q/rho_{i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="$\rho _{i}$">|; 

$key = q/displaystylerho_{i,j}^{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="$\displaystyle \rho _{i,j}^{max}$">|; 

$key = q/0lexle0.2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$0\le x \le 0.2$">|; 

$key = q/{eqnarraystar}F_{x}(rho_{i,j}^{n},rho_{i+1,j}^{n},u_{i+frac{1}{2},j}^{n})&=&[(u_|)rho_{i,j+1}^{n}]frac{Deltat}{2Deltay},{eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="636" HEIGHT="94" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="\begin{eqnarray*}
F_{x}(\rho _{i,j}^{n},\rho _{i+1,j}^{n},u_{i+\frac{1}{2},j}^{...
...}{2}}^{n}\vert)\rho _{i,j+1}^{n}]
\frac{\Delta t}{2\Delta y},
\end{eqnarray*}">|; 

$key = q/{displaymath}A_{i+frac{1}{2}}equivF_{i+frac{1}{2}}^{H}-F_{i+frac{1}{2}}^{L}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="\begin{displaymath}
A_{i+\frac{1}{2}}\equiv F_{i+\frac{1}{2}}^{H}-F_{i+\frac{1}{2}}^{L}.
\end{displaymath}">|; 

$key = q/Deltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$\Delta x$">|; 

$key = q/{displaymath}rho_{i}^{td}=rho_{i}^{n}-frac{1}{Deltax}[F_{i+frac{1}{2}}^{L}-F_{i-frac{1}{2}}^{L}].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="\begin{displaymath}
\rho _{i}^{td}=\rho _{i}^{n} - \frac{1}{\Delta x}[F_{i+\frac{1}{2}}^{L}
-F_{i-\frac{1}{2}}^{L}].
\end{displaymath}">|; 

$key = q/i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$i$">|; 

$key = q/displaystylembox{max}(0,A_{i+frac{1}{2},j})-mbox{min}(0,A_{i-frac{1}{2},j});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="$\displaystyle \mbox{max}(0, A_{i+\frac{1}{2},j}) - \mbox{min}(0,
A_{i-\frac{1}{2},j})$">|; 

$key = q/displaystylembox{max}(rho_{i}^{n},;rho_{i}^{td}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="$\displaystyle \mbox{max}(\rho _{i}^{n}, \; \rho _{i}^{td}),$">|; 

$key = q/{displaymath}frac{1}{2}left[rho_{2}+left(rho_{2}+frac{rho_{1}-rho_{2}}{Deltax+Deht)right]cdot(frac{1}{2}Deltax-u_{i}^{n}Deltat),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="558" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="\begin{displaymath}
\frac{1}{2}\left[\rho _{2} + \left(\rho _{2}+\frac{\rho _{1...
...ght)\right]\cdot
(\frac{1}{2}\Delta x - u_{i}^{n}\Delta t),
\end{displaymath}">|; 

$key = q/displaystyle+mbox{max}(0,A_{i,j-frac{1}{2}})-mbox{min}(0,A_{i,j+frac{1}{2}}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="$\displaystyle + \mbox{max}(0, A_{i,j-\frac{1}{2}}) -
\mbox{min}(0, A_{i,j+\frac{1}{2}}),$">|; 

$key = q/displaystyleC_{i,j+frac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="$\displaystyle C_{i,j+\frac{1}{2}}$">|; 

$key = q/{displaymath}overline{rho}_{i}^{n+1}=rho_{i}^{n+1}-(f^{c}_{i+frac{1}{2}}-f^{c}_{i-frac{1}{2}}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="\begin{displaymath}
\overline{\rho }_{i}^{n+1} = \rho _{i}^{n+1} -
(f^{c}_{i+\frac{1}{2}} - f^{c}_{i-\frac{1}{2}}).
\end{displaymath}">|; 

$key = q/displaystylerho_{i,j}^{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="$\displaystyle \rho _{i,j}^{b}$">|; 

$key = q/displaystyle(rho_{i,j}-rho_{i,j}^{min})DeltaxDeltay,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="$\displaystyle (\rho _{i,j}-\rho _{i,j}^{min})\Delta x \Delta y
,$">|; 

$key = q/{displaymath}left|ufrac{Deltat}{Deltax}right|<frac{1}{2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="\begin{displaymath}
\left\vert u \frac{\Delta t}{\Delta x}\right\vert &lt; \frac{1}{2},
\end{displaymath}">|; 

$key = q/displaystylembox{max}(0,A_{i+frac{1}{2}})-mbox{min}(0,A_{i-frac{1}{2}}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="$\displaystyle \mbox{max}(0, A_{i+\frac{1}{2}}) - \mbox{min}(0,
A_{i-\frac{1}{2}}),$">|; 

$key = q/rho_{i}^{n},rho_{i+1}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$\rho _ {i}^{n}, \rho _{i+1}^{n}$">|; 

$key = q/{displaymath}frac{1}{2}left[rho_{1}+left(rho_{2}+frac{rho_{1}-rho_{2}}{Deltax+De)right]cdot(frac{1}{2}Deltax+u_{i+1}^{n}Deltat),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="585" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="\begin{displaymath}
\frac{1}{2}\left[\rho _{1} + \left(\rho _{2}+\frac{\rho _{1...
...t)\right]\cdot
(\frac{1}{2}\Delta x + u_{i+1}^{n}\Delta t),
\end{displaymath}">|; 

$key = q/D_{i+frac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$D_{i+\frac{1}{2}}$">|; 

$key = q/displaystylembox{max}(rho_{i-1}^{a},;rho_{i}^{a},;rho_{i+1}^{a}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$\displaystyle \mbox{max}(\rho _{i-1}^{a}, \; \rho _
{i}^{a}, \; \rho _{i+1}^{a}),$">|; 

$key = q/O((u_{ipm1}-u_{i})DeltatslashDeltax)simO(epsilon)<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$O((u_{i\pm 1}-u_{i})\Delta t/\Delta x) \sim O(\epsilon ) &lt; 1$">|; 

$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="|."$dir".q|img190.png"
 ALT="$(x_{m}, y_{m})=(0.75, 0.5)$">|; 

$key = q/displaystylerho_{i,j}^{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="$\displaystyle \rho _{i,j}^{a}$">|; 

$key = q/{displaymath}rho_{i}^{n+1}=rho_{i}^{n}-frac{1}{2}epsilon(rho_{i+1}^{n}-rho_{i-1}on^{2}(rho_{i+1}^{n}-2rho_{i}^{n}+rho_{i-1}^{n}){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="\begin{displaymath}
\rho _{i}^{n+1} = \rho _{i}^{n} - \frac{1}{2}\epsilon (\rho...
...psilon ^{2}(\rho _{i+1}^{n}-2\rho _{i}
^{n}+\rho _{i-1}^{n})
\end{displaymath}">|; 

$key = q/displaystyleleft{array{{lcl}<mbox_mark>{min}(R_{i,j+1}^{+},R_{i,j}^{-})<col_markrk><mbox_mark>{if}<col_mark>A_{i,j+frac{1}{2}}<0.array{right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="$\displaystyle \left\{
\begin{array}{lcl}
\mbox{min}(R_{i,j+1}^{+}, R_{i,j}^{-})...
...}^{+}, R_{i,j+1}^{-}) &amp; \mbox{if} &amp;
A_{i,j+\frac{1}{2}} &lt; 0.
\end{array}\right.$">|; 

$key = q/{displaymath}tilde{rho}_{i}=rho_{i}^{n}-frac{Deltat}{Deltax}[rho_{i}^{n}u_{i}^{n}-rho_{i-1}^{n}u_{i-1}^{n}].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="\begin{displaymath}
\tilde{\rho }_{i} = \rho _{i}^{n}-\frac{\Delta t}{\Delta x}
[\rho _{i}^{n}u_{i}^{n}-\rho _{i-1}^{n}u_{i-1}^{n}].
\end{displaymath}">|; 

$key = q/{figure}center{{Depsf[80mm][]{ps-figslashfig2.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="209" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="\begin{figure}\begin{center}
\Depsf[80mm][]{ps-fig/fig2.ps}
\end{center} \end{figure}">|; 

$key = q/displaystylembox{min}(rho_{i}^{n},;rho_{i}^{td}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$\displaystyle \mbox{min}(\rho _{i}^{n}, \; \rho _{i}^{td}),$">|; 

$key = q/displaystylerho_{i,j}^{min};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="$\displaystyle \rho _{i,j}^{min}$">|; 

$key = q/{displaymath}DP{rho}{t}+DP{}{x}(rhou)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="\begin{displaymath}
\DP{\rho }{t} + \DP{}{x}(\rho u) = 0,
\end{displaymath}">|; 

$key = q/displaystyleP_{i,j}^{+};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$\displaystyle P_{i,j}^{+}$">|; 

$key = q/rho_{i}^{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$\rho _
{i}^{max}$">|; 

$key = q/displaystyleleft{array{{lcl}<mbox_mark>{min}(1,Q_{i,j}^{-}slashP_{i,j}^{-})<col_<col_mark><mbox_mark>{if}<col_mark>P_{i,j}^{-}=0.array{right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="$\displaystyle \left\{
\begin{array}{lcl}
\mbox{min}(1, Q_{i,j}^{-}/P_{i,j}^{-})...
...mbox{if} &amp; P_{i,j}^{-}&gt;0,
\\\\
0 &amp; \mbox{if} &amp; P_{i,j}^{-}=0.
\end{array}\right.$">|; 

$key = q/displaystyleoverline{rho}_{i}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$\displaystyle \overline{\rho }_{i}^{n+1}$">|; 

$key = q/sim;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$\sim $">|; 

$key = q/u_{i}^{n},u_{i+1}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$u_{i}^{n},
u_{i+1}^{n}$">|; 

$key = q/{displaymath}A_{i+frac{1}{2}}^{c}=C_{i+frac{1}{2}}A_{i+frac{1}{2}},;0leC_{i+frac{1}{2}}le1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="\begin{displaymath}
A_{i+\frac{1}{2}}^{c} = C_{i+\frac{1}{2}}A_{i+\frac{1}{2}}, \;
0 \le C_{i+\frac{1}{2}}\le 1.
\end{displaymath}">|; 

$key = q/displaystyleR_{i}^{+};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="$\displaystyle R_{i}^{+}$">|; 

$key = q/displaystyleR_{i,j}^{-};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="$\displaystyle R_{i,j}^{-}$">|; 

$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="|."$dir".q|img188.png"
 ALT="$\omega = 0.1$">|; 

$key = q/f_{ipmfrac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$f_{i\pm \frac{1}{2}}$">|; 

$key = q/textstylembox{if};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$\textstyle \mbox{if}$">|; 

$key = q/displaystyleA_{i+frac{1}{2}}(rho_{i}^{td}-rho_{i-1}^{td})<0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$\displaystyle A_{i+\frac{1}{2}}(\rho _
{i}^{td} - \rho _{i-1}^{td}) &lt; 0.$">|; 

$key = q/displaystyleA_{i,j+frac{1}{2}}(rho_{i,j}^{td}-rho_{i,j-1}^{td})<0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="$\displaystyle A_{i,j+\frac{1}{2}}(\rho _
{i,j}^{td} - \rho _{i,j-1}^{td}) &lt; 0.$">|; 

$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="|."$dir".q|img187.png"
 ALT="$(x_{0}, y_{0})=(0.5, 0.5)$">|; 

$key = q/displaystylembox{min}(rho_{i-1,j}^{td},;rho_{i,j}^{td},;rho_{i+1,j}^{td},;rho_{i,j-1}^{td},;rho_{i,j+1}^{td}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="$\displaystyle \mbox{min}(\rho _{i-1,j}^{td}, \; \rho _
{i,j}^{td}, \; \rho _{i+1,j}^{td}, \; \rho _{i,j-1}^{td}, \;
\rho _{i,j+1}^{td}).$">|; 

$key = q/displaystyleC_{i+frac{1}{2},j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="$\displaystyle C_{i+\frac{1}{2},j}$">|; 

$key = q/overline{rho}_{i}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$\overline{\rho }_
{i}^{n+1}$">|; 

$key = q/rho_{i}^{min};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$\rho _{i}^{min}$">|; 

$key = q/{eqnarraystar}rho_{i}^{n+1}&=&frac{1}{2}Q_{-}^{2}(rho_{i-1}^{n}-rho_{i}^{n})+frai}^{n})frac{Deltat}{Deltax}right.right].{eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="497" HEIGHT="91" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="\begin{eqnarray*}
\rho _{i}^{n+1} &amp;=&amp; \frac{1}{2}Q_{-}^{2}(\rho _
{i-1}^{n}-\r...
...m 1}^{n} - u_{i}^{n}) \frac{\Delta
t}{\Delta x}\right.\right].
\end{eqnarray*}">|; 

$key = q/{displaymath}overline{rho}_{i}^{n+1}=rho_{i}^{n+1}-frac{1}{8}(rho_{i+1}^{n+1}-2rho_{i}^{n+1}+rho_{i-1}^{n+1}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="\begin{displaymath}
\overline{\rho }_{i}^{n+1} = \rho _{i}^{n+1}
- \frac{1}{8}(\rho _{i+1}^{n+1}-2\rho _{i}^{n+1}
+\rho _{i-1}^{n+1}),
\end{displaymath}">|; 

$key = q/displaystyle-[C_{i,j-frac{1}{2}}G_{i,j-frac{1}{2}}^{H}+(1-C_{i,j-frac{1}{2}})G_{i,j-frac{1}{2}}^{L}]},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="$\displaystyle - [C_{i,j-\frac{1}{2}}G_{i,j-\frac{1}{2}}^{H}+
(1-C_{i,j-\frac{1}{2}})G_{i,j-\frac{1}{2}}^{L}]\},$">|; 

$key = q/displaystyleA_{i+frac{1}{2},j}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="$\displaystyle A_{i+\frac{1}{2},j} =0$">|; 

$key = q/displaystyleleft(frac{1}{2}mpu_{i}^{n}frac{Deltat}{Deltax}right)leftslashleft[1pm(u_{ipm1}^{n}-u_{i}^{n})frac{Deltat}{Deltax}right.right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$\displaystyle \left(\frac{1}{2} \mp u_{i}^{n}\frac{\Delta t}{\Delta x}
\right)\...
...eft[1 \pm (u_{i\pm 1}^{n} - u_{i}^{n})
\frac{\Delta t}{\Delta x}\right.\right],$">|; 

$key = q/rho_{i+1}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$\rho _
{i+1}^{n+1}$">|; 

$key = q/{displaymath}frac{1}{2}left[rho_{2}+left(rho_{3}+frac{rho_{2}-rho_{3}}{Deltax+Deht)right]cdot(frac{1}{2}Deltax+u_{i}^{n}Deltat),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="557" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="\begin{displaymath}
\frac{1}{2}\left[\rho _{2} + \left(\rho _{3}+\frac{\rho _{2...
...ght)\right]\cdot
(\frac{1}{2}\Delta x + u_{i}^{n}\Delta t),
\end{displaymath}">|; 

$key = q/textstylembox{andeither};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$\textstyle \mbox{and either}$">|; 

$key = q/{figure}center{{Depsf[][]{ps-figslashfig6.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][]{ps-fig/fig6.ps}
\end{center} \end{figure}">|; 

$key = q/0lexle1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$0\le x \le 1$">|; 

$key = q/C_{ipmfrac{1}{2},j},C_{i,jpmfrac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="$C_{i\pm \frac{1}{2},j}, C_{i,j\pm \frac{1}{2}}$">|; 

$key = q/{displaymath}S=left{array{{lcl}+1&mbox{if}&Delta_{i+frac{1}{2}}geq0,-1&mbox{if}&Deltaf_{i+frac{1}{2}}<0.array{right.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="\begin{displaymath}
S = \left\{
\begin{array}{lcl}
+1 &amp; \mbox{if} &amp; \Delta _...
...box{if} &amp; \Delta f_{i+\frac{1}{2}} &lt; 0.
\end{array} \right.
\end{displaymath}">|; 

$key = q/textstylembox{or};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$\textstyle \mbox{or}$">|; 

$key = q/displaystyleA_{i+frac{1}{2}}(rho_{i+1}^{td}-rho_{i}^{td})<0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="$\displaystyle A_{i+\frac{1}{2}}(\rho _
{i+1}^{td} - \rho _{i}^{td}) &lt; 0,$">|; 

$key = q/rho_{i}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$\rho _{i}^{n+1}$">|; 

$key = q/{displaymath}f^{c}_{i+frac{1}{2}}=S_{i+frac{1}{2}}mbox{max}{0,mbox{min}(S_{i+fra1}{2}}|,;S_{i+frac{1}{2}}Delta_{i+frac{3}{2}})},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="\begin{displaymath}
f^{c}_{i+\frac{1}{2}} = S_{i+\frac{1}{2}} \mbox{max} \{0,
...
...}{2}}\vert, \; S_{i+\frac{1}{2}}\Delta _
{i+\frac{3}{2}})\},
\end{displaymath}">|; 

$key = q/{displaymath}S=left{array{{lcl}+1&mbox{if}&f_{i+frac{1}{2}}geq0-1&mbox{if}&f_{i+frac{1}{2}}<0array{right.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="\begin{displaymath}
S = \left\{
\begin{array}{lcl}
+1 &amp; \mbox{if} &amp; f_{i+\fr...
...
-1 &amp; \mbox{if} &amp; f_{i+\frac{1}{2}} &lt; 0
\end{array} \right.
\end{displaymath}">|; 

$key = q/R_{i}^{-};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$R_{i}^{-}$">|; 

$key = q/displaystyle(rho_{i}^{max}-rho_{i})Deltax,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$\displaystyle (\rho _{i}^{max}-\rho _{i})\Delta x,$">|; 

$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|img197.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][80mm]{ps-fig/upstream1.ps}
\Depsf[][80mm]{ps-fig/upstream2.ps}
\end{center} \end{figure}">|; 

$key = q/displaystylembox{max}(0,A_{i-frac{1}{2}})-mbox{min}(0,A_{i+frac{1}{2}}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$\displaystyle \mbox{max}(0, A_{i-\frac{1}{2}}) - \mbox{min}(0,
A_{i+\frac{1}{2}}),$">|; 

$key = q/displaystylerho_{i}^{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$\displaystyle \rho _{i}^{max}$">|; 

$key = q/Delta_{i-frac{1}{2}},Delta_{i+frac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$\Delta _
{i-\frac{1}{2}}, \Delta _{i+\frac{1}{2}}$">|; 

$key = q/displaystyleleft{array{{lcl}<mbox_mark>{min}(1,Q_{i}^{+}slashP_{i}^{+})<col_mark>0<col_mark><mbox_mark>{if}<col_mark>P_{i}^{+}=0.array{right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$\displaystyle \left\{
\begin{array}{lcl}
\mbox{min}(1, Q_{i}^{+}/P_{i}^{+}) &amp; \mbox{if} &amp; P_{i}^{+}&gt;0, \\\\
0 &amp; \mbox{if} &amp; P_{i}^{+}=0.
\end{array}\right.$">|; 

$key = q/displaystyleQ_{i}^{+};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="$\displaystyle Q_{i}^{+}$">|; 

$key = q/displaystylef_{ipmfrac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$\displaystyle f_{i\pm \frac{1}{2}}$">|; 

$key = q/displaystyleQ_{pm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\displaystyle Q_{\pm }$">|; 

$key = q/displaystylerho_{i}^{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$\displaystyle \rho _{i}^{a}$">|; 

$key = q/{displaymath}rho_{i}^{td}=rho_{i}^{n}-frac{1}{Deltax}[A_{i+frac{1}{2}}^{c}-A_{i-frac{1}{2}}^{c}].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="\begin{displaymath}
\rho _{i}^{td}=\rho _{i}^{n} - \frac{1}{\Delta x}[A_{i+\frac{1}{2}}^{c}
-A_{i-\frac{1}{2}}^{c}].
\end{displaymath}">|; 

$key = q/displaystylembox{格子点{i,j{から出て行く全てのantidiffusiveフラックスの和};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="501" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="$\displaystyle \mbox{格子点 $i,j$ から出て行く全ての antidiffusive
フラックスの和}$">|; 

$key = q/i+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$i+1$">|; 

$key = q/(uDeltatslashDeltax);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="$(u\Delta t/\Delta x)$">|; 

$key = q/displaystylembox{max}(rho_{i-1,j}^{td},;rho_{i,j}^{td},;rho_{i+1,j}^{td},;rho_{i,j-1}^{td},;rho_{i,j+1}^{td}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="$\displaystyle \mbox{max}(\rho _{i-1,j}^{td}, \; \rho _
{i,j}^{td}, \; \rho _{i+1,j}^{td}, \; \rho _{i,j-1}^{td}, \;
\rho _{i,j+1}^{td}),$">|; 

$key = q/rho_{1},rho_{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$\rho _{1}, \rho _{2}$">|; 

$key = q/displaystylembox{格子点{i{から出て行く全てのantidiffusiveフラックスの和};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="484" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$\displaystyle \mbox{格子点 $i$ から出て行く全ての antidiffusive フ
ラックスの和}$">|; 

$key = q/overline{rho}=sqrt{rho_{min}rho_{max}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$\overline{\rho }=\sqrt{\rho _{min}\rho_{max}}$">|; 

$key = q/displaystylembox{max}(rho_{i-1,j}^{a},;rho_{i,j}^{a},;rho_{i+1,j}^{a},;rho_{i,j-1}^{a},;rho_{i,j+1}^{a}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$\displaystyle \mbox{max}(\rho _{i-1,j}^{a}, \; \rho _ {i,j}^{a}
, \; \rho _{i+1,j}^{a}, \; \rho _{i,j-1}^{a}, \;
\rho _{i,j+1}^{a}),$">|; 

$key = q/O(epsilon);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$O(\epsilon )$">|; 

$key = q/textstyleequiv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$\textstyle \equiv$">|; 

$key = q/displaystyleP_{i,j}^{-};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$\displaystyle P_{i,j}^{-}$">|; 

$key = q/{displaymath}epsilon_{i+frac{1}{2}}=frac{Deltat}{Deltax}mbox{max}[u_{i}^{n},u_{i+1}^{n}].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="\begin{displaymath}
\epsilon _{i+\frac{1}{2}} = \frac{\Delta t}{\Delta x}
\mbox{max}[u_{i}^{n}, u_{i+1}^{n}].
\end{displaymath}">|; 

$key = q/O((u_{ipm1}-u_{i})DeltatslashDeltax)<O(epsilon);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$O((u_{i\pm 1}-u_{i})\Delta t/\Delta x) &lt; O(\epsilon )$">|; 

$key = q/displaystylefrac{rho_{i+1}^{n}Deltax}{Deltax+Deltat(u_{i+1}^{n}-u_{i}^{n})},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$\displaystyle \frac{\rho _{i+1}^{n}\Delta x}{\Delta x + \Delta t
(u_{i+1}^{n} - u_{i}^{n})},$">|; 

$key = q/{figure}center{{Depsf[][80mm]{ps-figslashfct-exp1.ps}{{{{Depsf[][80mm]{ps-figslashfct-exp2.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="726" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][80mm]{ps-fig/fct-exp1.ps}
\Depsf[][80mm]{ps-fig/fct-exp2.ps}
\end{center} \end{figure}">|; 

$key = q/(x,y)=(iDeltax,jDeltay);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$(x,y)=(i\Delta x, j\Delta y)$">|; 

$key = q/displaystylembox{max}(rho_{i,j}^{n},;rho_{i,j}^{td}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="$\displaystyle \mbox{max}(\rho _{i,j}^{n}, \; \rho _
{i,j}^{td}),$">|; 

$key = q/{figure}center{{Depsf[][80mm]{ps-figslashinit-cylind.ps}{{{{Depsf[][80mm]{ps-figslashfct-exp3.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="726" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][80mm]{ps-fig/init-cylind.ps}
\Depsf[][80mm]{ps-fig/fct-exp3.ps}
\end{center} \end{figure}">|; 

$key = q/rho=2.0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$\rho =2.0$">|; 

$key = q/{displaymath}S=left{array{{lcl}+1&mbox{if}&f_{i+frac{1}{2}}geq0,-1&mbox{if}&f_{i+frac{1}{2}}<0.array{right.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="\begin{displaymath}
S = \left\{
\begin{array}{lcl}
+1 &amp; \mbox{if} &amp; f_{i+\fr...
...-1 &amp; \mbox{if} &amp; f_{i+\frac{1}{2}} &lt; 0.
\end{array} \right.
\end{displaymath}">|; 

$key = q/displaystylerho_{i}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$\displaystyle \rho _{i}^{n+1}$">|; 

$key = q/{figure}center{{Depsf[][]{ps-figslashfig5.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="445" HEIGHT="430" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][]{ps-fig/fig5.ps}
\end{center} \end{figure}">|; 

$key = q/epsilon=uDeltatslashDeltax=0.2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$\epsilon = u\Delta t/\Delta x=0.2$">|; 

$key = q/y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$y$">|; 

$key = q/{displaymath}frac{1}{4}(u_{i-1}^{n}-u_{i}^{n})(rho_{i-1}^{n}+rho_{i}^{n})frac{De)(rho_{i+1}^{n}+rho_{i}^{n})frac{Deltat}{Deltax}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="465" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="\begin{displaymath}
\frac{1}{4}(u_{i-1}^{n}-u_{i}^{n})(\rho _{i-1}^{n} + \rho _...
...
(\rho _{i+1}^{n} + \rho _ {i}^{n})\frac{\Delta t}{\Delta x}
\end{displaymath}">|; 

$key = q/rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$\rho $">|; 

$key = q/Deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$\Delta
t$">|; 

$key = q/rho_{i+frac{1}{2}}^{n+frac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$\rho _
{i+\frac{1}{2}}^{n+\frac{1}{2}}$">|; 

$key = q/displaystylerho_{i.j}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="$\displaystyle \rho _{i.j}^{n+1}$">|; 

$key = q/{figure}center{{Depsf[][80mm]{ps-figslashinit-cone.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="363" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][80mm]{ps-fig/init-cone.ps}
\end{center} \end{figure}">|; 

$key = q/displaystyleleft{array{{lcl}<mbox_mark>{min}(1,Q_{i}^{-}slashP_{i}^{-})<col_mark>0<col_mark><mbox_mark>{if}<col_mark>P_{i}^{-}=0.array{right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="$\displaystyle \left\{
\begin{array}{lcl}
\mbox{min}(1, Q_{i}^{-}/P_{i}^{-}) &amp; \mbox{if} &amp; P_{i}^{-}&gt;0, \\\\
0 &amp; \mbox{if} &amp; P_{i}^{-}=0.
\end{array}\right.$">|; 

$key = q/displaystyleA_{i+frac{1}{2}}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$\displaystyle A_{i+\frac{1}{2}} =0$">|; 

$key = q/A_{i+frac{1}{2}}(rho_{i+1}^{td}-rho_{i}^{td})<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="$ A_{i+\frac{1}{2}}(\rho _ {i+1}^{td} - \rho _
{i}^{td}) &lt; 0 $">|; 

$key = q/displaystyle+[C_{i,j+frac{1}{2}}G_{i,j+frac{1}{2}}^{H}+(1-C_{i,j+frac{1}{2}})G_{i,j+frac{1}{2}}^{L}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="$\displaystyle + [C_{i,j+\frac{1}{2}}G_{i,j+\frac{1}{2}}^{H}+
(1-C_{i,j+\frac{1}{2}})G_{i,j+\frac{1}{2}}^{L}]$">|; 

$key = q/displaystyleP_{i}^{-};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="$\displaystyle P_{i}^{-}$">|; 

$key = q/x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$x$">|; 

$key = q/mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$\mu $">|; 

$key = q/displaystylerho_{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$\displaystyle \rho _{2}$">|; 

$key = q/{eqnarraystar}&&left(frac{1}{2}+epsilonright)^{2}(u_{i-1}^{n}-u_{i}^{n})(rho_{i-})right]frac{Deltat}{Deltax}rho_{i}^{n},{eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="626" HEIGHT="96" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="\begin{eqnarray*}
&amp;&amp; \left(\frac{1}{2}+\epsilon \right)
^{2}(u_{i-1}^{n}-u_{i}...
...^{n}-u_{i}^{n})
\right]\frac{\Delta t}{\Delta x}\rho _{i}^{n},
\end{eqnarray*}">|; 

$key = q/displaystyleleft{array{{lcl}<mbox_mark>{min}(R_{i+1,j}^{+},R_{i,j}^{-})<col_markrk><mbox_mark>{if}<col_mark>A_{i+frac{1}{2},j}<0.array{right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="$\displaystyle \left\{
\begin{array}{lcl}
\mbox{min}(R_{i+1,j}^{+}, R_{i,j}^{-})...
...}^{+}, R_{i+1,j}^{-}) &amp; \mbox{if} &amp;
A_{i+\frac{1}{2},j} &lt; 0.
\end{array}\right.$">|; 

$key = q/displaystyle(rho_{i,j}^{max}-rho_{i,j})DeltaxDeltay,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="$\displaystyle (\rho _{i,j}^{max}-\rho _{i,j})\Delta x \Delta y,$">|; 

$key = q/F_{i+frac{1}{2}}^{L},F_{i+frac{1}{2}}^{H};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$F_{i+\frac{1}{2}}^{L}, F_{i+\frac{1}{2}}^{H}$">|; 

$key = q/epsilonequivu_{i}^{n}DeltatslashDeltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$\epsilon \equiv u_{i}^{n}\Delta t/\Delta x$">|; 

$key = q/{displaymath}C_{i+frac{1}{2}}=left{array{{lcl}mbox{min}(R_{i+1}^{+},R_{i}^{-})&m1}^{-})&mbox{if}&A_{i+frac{1}{2}}<0.array{right.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="\begin{displaymath}
C_{i+\frac{1}{2}} = \left\{
\begin{array}{lcl}
\mbox{min}...
...}) &amp; \mbox{if} &amp;
A_{i+\frac{1}{2}} &lt; 0.
\end{array} \right.
\end{displaymath}">|; 

$key = q/displaystylembox{max}(0,A_{i-frac{1}{2},j})-mbox{min}(0,A_{i+frac{1}{2},j});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="$\displaystyle \mbox{max}(0, A_{i-\frac{1}{2},j}) - \mbox{min}(0,
A_{i+\frac{1}{2},j})$">|; 

$key = q/displaystylerho_{i}^{min};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="$\displaystyle \rho _{i}^{min}$">|; 

$key = q/f^{c}_{ipmfrac{1}{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$f^{c}_{i\pm
\frac{1}{2}}$">|; 

$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="|."$dir".q|img186.png"
 ALT="$\Delta x =
\Delta y = 0.01$">|; 

$key = q/rho_{i}^{max},rho_{i}^{min};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="$\rho _{i}^{max}, \rho _{i}^{min}$">|; 

$key = q/{displaymath}mu=left{array{{ll}0.125[1+(1-0.25overline{rho}slash|D_{i+frac{1}{2}5overline{rho}0.125&mbox{otherwise}.array{right.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="444" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="\begin{displaymath}
\mu = \left\{
\begin{array}{ll}
0.125[1+(1-0.25\overline{...
...ne{\rho } \\\\
0.125 &amp; \mbox{otherwise}.
\end{array} \right.
\end{displaymath}">|; 

$key = q/f_{i}^{n}=u_{i}^{n}rho_{i}^{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$f_{i}^{n}=u_{i}^{n}\rho _{i}^{n}$">|; 

$key = q/displaystyleA_{i,j+frac{1}{2}}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="$\displaystyle A_{i,j+\frac{1}{2}} =0$">|; 

$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="|."$dir".q|img189.png"
 ALT="$(u, v) = ( - \omega (y-y_{0}),
\omega (x-x_{0}))$">|; 

$key = q/{figure}center{{Depsf[][20cm]{ps-figslashfig4.ps}{{{center{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="639" HEIGHT="885" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="\begin{figure}\begin{center}
\Depsf[][20cm]{ps-fig/fig4.ps}
\end{center} \end{figure}">|; 

$key = q/(iDeltax,jDeltay);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$(i\Delta x, j\Delta y)$">|; 

$key = q/displaystylefrac{rho_{i}^{n}Deltax}{Deltax+Deltat(u_{i+1}^{n}-u_{i}^{n})},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$\displaystyle \frac{\rho _{i}^{n}\Delta x}{\Delta x + \Delta t
(u_{i+1}^{n} - u_{i}^{n})},$">|; 

$key = q/{displaymath}Delta_{i+frac{1}{2}}=tilde{rho}_{i+1}-tilde{rho}_{i}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="\begin{displaymath}
\Delta _{i+\frac{1}{2}} = \tilde{\rho }_{i+1}-\tilde{\rho }_{i}.
\end{displaymath}">|; 

$key = q/R_{i}^{+};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="$R_{i}^{+}$">|; 

$key = q/displaystylerho_{i,j}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="$\displaystyle \rho _{i,j}^{n+1}$">|; 

$key = q/displaystylembox{max}(rho_{i-1}^{td},;rho_{i}^{td},;rho_{i+1}^{td}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="$\displaystyle \mbox{max}(\rho _{i-1}^{td}, \; \rho _
{i}^{td}, \; \rho _{i+1}^{td}),$">|; 

$key = q/displaystyleR_{i,j}^{+};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="$\displaystyle R_{i,j}^{+}$">|; 

$key = q/{eqnarraystar}rho_{i+frac{1}{2}}^{n+frac{1}{2}}&=&frac{1}{2}(rho_{i}^{n}+rho_{i+{1}{2}}-f_{i-frac{1}{2}}^{n+frac{1}{2}}){eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="90" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="\begin{eqnarray*}
\rho _{i+\frac{1}{2}}^{n+\frac{1}{2}} &amp;=&amp; \frac{1}{2}(\rho _{...
...\frac{1}{2}}^{n+\frac{1}{2}}-f_{i-\frac{1}{2}}^{n+\frac{1}{2}})
\end{eqnarray*}">|; 

$key = q/i-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$i-1$">|; 

1;