% $BI=Bj(B  $B0\N.J}Dx<0$N:9J,2rK!(B: FCT $BK!(B  Zalesak(1979)$B$K$h$k(BFCT
%
% $BMzNr(B  1998/01/19 $B>.9b@5;L(B
%
\section{Zalesak(1979) $B$K$h$kI=8=(B: 1$B<!85(B}

    Zalesak(1979) $B$O$h$j0lHL2=$5$l$?(B FCT $B$NI=8=$rDs0F$7$?(B. $B8=:_(B FCT $B$H(B
    $B$$$&$H$-$K$O(B Zalesak(1979) $B$K$h$kDj5A$r;X$9$N$,IaDL$G$"$k(B. $B$3$NDj(B
    $B5A$K=>$&$HA4@a$G5s$2$?$$$/$D$+$NLdBjE@$r$$$/$i$+2r7h$9$k$3$H$,$G$-(B
    $B$k(B. $B0J2<$G$O$3$N(B Zalesak(1979) $B$K$h$C$FM?$($i$l$?(B FCT $B$K$D$$$F8+$F(B
    $B$$$/$3$H$K$9$k(B.

    $B4JC1$N$?$a2r$/<0$O(B1$B<!85$N0\N.J}Dx<0(B\Deqref{eq: 1}$B$H$9$k(B. $B$3$l$rJ](B
    $BB87A$G:9J,2=$9$k$H(B,

    \begin{equation}
      \rho _{i}^{n+1}=\rho _{i}^{n} - \frac{1}{\Delta x}[F_{i+\frac{1}{2}}
      -F_{i-\frac{1}{2}}], \Deqlab{eq: 13}
    \end{equation}

    $B$H$J$k(B. $B$3$l$K(B FCT $B$rE,MQ$9$k$3$H$r9M$($k(B. 

    \subsection{$B7W;;<j=g(B}

    Zalesak(1979) $B$K=>$&$H$=$N7W;;<j=g$O0J2<$N$h$&$K$J$k(B.

    \begin{enumerate}
    \item $B@:EY$NDc$$Dc<!$N%9%-!<%`$rMQ$$$F%U%i%C%/%9$r7W;;$9$k(B. $B$3$l$r(B
      $F_{i+\frac{1}{2}}^{L}$ $B$HI=$9(B. $B$3$N$H$-MQ$$$k%9%-!<%`$O>eN.:9J,(B
      $B$N$h$&$J!VC1D4$J!W(B($B?tCMGH$r:n$i$J$$(B)$B%9%-!<%`$G$"$k$3$H$,MW@A$5$l(B
      $B$k(B.
    \item $B@:EY$N9b$$9b<!$N%9%-!<%`$rMQ$$$F%U%i%C%/%9$r7W;;$9$k(B. $B$3$l$r(B
      $F_{i+\frac{1}{2}}^{H}$ $B$HI=$9(B. 
    \item antidiffusive $B%U%i%C%/%9$r0J2<$N$h$&$K$7$FDj5A$9$k(B.
      \begin{equation}
        A_{i+\frac{1}{2}}\equiv F_{i+\frac{1}{2}}^{H}-F_{i+\frac{1}{2}}^{L}.
        \Deqlab{eq: 14}
      \end{equation}
    \item $BDc<!$N%9%-!<%`$G7W;;$9$k(B.
      \begin{equation}
        \rho _{i}^{td}=\rho _{i}^{n} - \frac{1}{\Delta x}[F_{i+\frac{1}{2}}^{L}
        -F_{i-\frac{1}{2}}^{L}]. \Deqlab{eq: 15}
      \end{equation}
    \item $BJd@5%U%i%C%/%9$rE,Ev$JJ}K!$G7W;;$9$k(B. $BJd@5$NM?$($k:]$N>r7o$O(B
      Boris and Book(1973) $B$HF1MM$G$"$k(B.
      \begin{equation}
        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. \Deqlab{eq: 16}
      \end{equation}

      $B$J$*Jd@578?t(B $C_{i+\frac{1}{2}}$ $B$NM?$(J}$K$D$$$F$O8e=R$9$k(B.
    \item $BJd@5%U%i%C%/%9$rM?$($k(B.
      \begin{equation}
        \rho _{i}^{td}=\rho _{i}^{n} - \frac{1}{\Delta x}[A_{i+\frac{1}{2}}^{c}
        -A_{i-\frac{1}{2}}^{c}]. \Deqlab{eq: 17}
      \end{equation}
    \end{enumerate}

    $B$3$N$h$&$KDj5A$9$k$3$H$K$h$j1~MQHO0O$,Bg$-$/9-$,$k(B. $BDc<!$N%9%-!<%`(B
    $B$K$O>r7o$,$D$/$,(B, $B9b<!$N%9%-!<%`$KBP$7$F$O2?$b@)Ls$,$J$$$N$G$"$i$f(B
    $B$k%9%-!<%`$rMQ$$$k$3$H$,$G$-$k(B. $BAH$_9g$o$;$k%9%-!<%`$r7h$a$F$7$^$((B
    $B$P$"$H$O>e5-$N<j=g$K1h$C$F5!3#E*$K7W;;$r?J$a$k$3$H$,$G$-$k(B. $B<B:]7W(B
    $B;;$r$9$k>l9g$K$O(B $F_{i+\frac{1}{2}}^{L}, F_{i+\frac{1}{2}}^{H}$ $B$r(B
    $B7W;;$9$kItJ,$N%3!<%I$@$1$r=q$-49$($l$P$h$$$3$H$K$J$k$N$G(B, $B<o!9$N%9(B
    $B%-!<%`$rMQ$$$F$=$N%Q%U%)!<%^%s%9$rHf3S8!F$$9$k$3$H$,MF0W$K$J$k(B.

    $B$J$*(B Harten and Zwas(1972) $B$K$h$k<+8JD4@a:.9g%9%-!<%`(B
    (Self-Adjusting Hybrid Schemes; $B0J2<(B SAHS $B$H$9$k(B)$B$O(B, $B>e5-$N(B 5 $BHV(B
    $BL\$KAjEv$9$k=j$N7W;;J}K!$,0[$J$k$@$1$G(B, $B$=$NB>$N<j=g$OF1MM$G$"$k(B. 

    \Deqref{eq: 14}$\sim $\Deqref{eq: 17}$B$r$^$H$a$F=q$/$3$H$b$G$-$k(B.
    $B$3$N>l9g(B

    \begin{screen}
    \begin{equation}
      \rho _{i}^{n+1}=\rho _{i}^{n}
       - \frac{1}{\Delta x}\left\{[C_{i+\frac{1}{2}}F_{i+\frac{1}{2}}^{H}+
      (1-C_{i+\frac{1}{2}})F_{i+\frac{1}{2}}^{L}]
      - [C_{i-\frac{1}{2}}F_{i-\frac{1}{2}}^{H}+
      (1-C_{i-\frac{1}{2}})F_{i-\frac{1}{2}}^{L}]\right\}, \Deqlab{eq: 18}
    \end{equation}
    \end{screen}
    
    $B$H$J$k(B. FCT $B$H$ODc<!$H9b<!$N%9%-!<%`$N=E$M9g$o$;$G$"$j(B, $B$=$NHfN((B
    $C_{i+\frac{1}{2}}$ $B$NM?$(J}$K9)IW$r6E$i$7$?$b$N$HB*$($k$3$H$,$G$-(B
    $B$k(B. $C_{i+\frac{1}{2}}$ $B$NM?$(J}$rJQ$($k$H(B SAHS $B$K$9$k$3$H$b$G$-$k(B.
    $B0lHL$K(B FCT $B$H$7$FIaDLL\$K$9$k$N$O(B\Deqref{eq: 18}$B<0$NI=8=$G$"$k(B.
    $B$3$3$^$GMh$k$H(B Boris and Book(1973), Book {\it et al.}(1975) $B$NI=(B
    $B8=$H$O0l8+0[$J$k$,(B, $B$d$C$F$$$k$3$H$OF1$8$G$"$k(B.
    
    \subsection{$BJd@578?t$N7h$aJ}(B}

    $BJd@578?t(B $C_{i+\frac{1}{2}}$ $B$OJd@5%U%i%C%/%9$K$h$C$FJd@5$5$l$?CM(B
    $B$,6KCM$r<h$i$J$$$h$&$K(B, $B$9$J$o$A(B $\rho _{i}$ $B$,$"$k:GBgCM(B $\rho _
    {i}^{max}$ $B5Z$S(B $\rho _{i}^{min}$ $B$rD6$($J$$$h$&$KM?$($k(B. $B$=$N:]BP(B
    $B>]$H$J$k3J;RE@$K4X$o$kA4$F$NJd@5%U%i%C%/%9$NB8:_$r9MN8$9$l$P(B, $BA4@a(B
    $B$G5s$2$?$h$&$JB?<!85$K3HD%$7$?>l9g$KH/@8$9$kLdBj$r2sHr$9$k$3$H$,$G(B
    $B$-$k$@$m$&(B. $B$3$l$O<!$N$h$&$J<j=g$G(B $C_{i+\frac{1}{2}}$ $B$rM?$($k$3(B
    $B$H$GC#@.$9$k$3$H$,$G$-$k(B.

    \begin{enumerate}
      \item antidifuusive $B%U%i%C%/%9$KBP$7$"$i$+$8$a0J2<$N>r7o$r2]$7$F$*$/(B.
        \begin{eqnarray}
          A_{i+\frac{1}{2}} =0 & \mbox{if} & A_{i+\frac{1}{2}}(\rho _
          {i+1}^{td} - \rho _{i}^{td}) < 0, \nonumber \\
          & \mbox{and either} & A_{i+\frac{1}{2}}(\rho _
          {i+2}^{td} - \rho _{i+1}^{td}) < 0, \nonumber \\
          & \mbox{or} & A_{i+\frac{1}{2}}(\rho _
          {i}^{td} - \rho _{i-1}^{td}) < 0. \Deqlab{eq: 19}
        \end{eqnarray}

        $B$3$l$O(B Book {\it et al.}(1975) $B$G8@$&$H$3$m$N(B ``terrace'' $B$r:n(B
        $B$i$J$$$h$&$K$9$k$?$a$N$b$N$G$"$k(B(Book {\it et al.} 1975, $B?^(B9$B;2(B
        $B>H(B). $B<B:]$O(B $ A_{i+\frac{1}{2}}(\rho _ {i+1}^{td} - \rho _
        {i}^{td}) < 0 $ $B$H$J$k$3$H$O5)$J$N$G(B, $B$3$N>r7o$rM?$($J$/$F$b$5(B
        $B$[$ILdBj$K$O$J$i$J$$(B.

      \item $\rho _{i}^{max}, \rho _{i}^{min}$ $B$r0J2<$N$h$&$K$7$FM?$($k(B.
        \begin{eqnarray}
          \rho _{i}^{max} &=& \mbox{max}(\rho _{i-1}^{td}, \; \rho _
          {i}^{td}, \; \rho _{i+1}^{td}), \Deqlab{eq: 20} \\
          \rho _{i}^{min} &=& \mbox{min}(\rho _{i-1}^{td}, \; \rho _
          {i}^{td}, \; \rho _{i+1}^{td}). \Deqlab{eq: 21}
        \end{eqnarray}

        $B$^$?$O(B clipping $B$rM^$($k$?$a$K(B, 
        \begin{eqnarray}
          \rho _{i}^{a} &=& \mbox{max}(\rho _{i}^{n}, \; \rho _{i}^{td}), 
          \Deqlab{eq: 22} \\
          \rho _{i}^{max} &=& \mbox{max}(\rho _{i-1}^{a}, \; \rho _
          {i}^{a}, \; \rho _{i+1}^{a}), \Deqlab{eq: 23} \\
          \rho _{i}^{b} &=& \mbox{min}(\rho _{i}^{n}, \; \rho _{i}^{td}), 
          \Deqlab{eq: 24} \\
          \rho _{i}^{min} &=& \mbox{min}(\rho _{i-1}^{b}, \; \rho _
          {i}^{b}, \; \rho _{i+1}^{b}), \Deqlab{eq: 25}
        \end{eqnarray}

        $B$H$9$k(B.

      \item $B0J2<$N(B3$B$D$NNL$rMQ0U$9$k(B.
        \begin{eqnarray}
          P_{i}^{+} &=& \mbox{$B3J;RE@(B $i$ $B$KF~$kA4$F$N(B antidiffusive $B%U(B
          $B%i%C%/%9$NOB(B} \nonumber \\ 
          &=& \mbox{max}(0, A_{i-\frac{1}{2}}) - \mbox{min}(0,
          A_{i+\frac{1}{2}}), \Deqlab{eq: 26} \\
          Q_{i}^{+} &=& (\rho _{i}^{max}-\rho _{i})\Delta x, \Deqlab{eq: 27} \\
          R_{i}^{+} &=& \left\{
          \begin{array}{lcl}
            \mbox{min}(1, Q_{i}^{+}/P_{i}^{+}) & \mbox{if} & P_{i}^{+}>0, \\
            0 & \mbox{if} & P_{i}^{+}=0.
          \end{array}
          \right. \Deqlab{eq: 28}
        \end{eqnarray}

        $R_{i}^{+}$ $B$O(B $i$ $BE@$KN.F~$7$&$k(B antidifuusive $B%U%i%C%/%9$N>e(B
        $B8B$r7h$a$k(B. 

      \item $BF1MM$K0J2<$N(B3$B$D$NNL$rMQ0U$9$k(B.
        \begin{eqnarray}
          P_{i}^{-} &=& \mbox{$B3J;RE@(B $i$ $B$+$i=P$F9T$/A4$F$N(B antidiffusive $B%U(B
          $B%i%C%/%9$NOB(B} \nonumber \\ 
          &=& \mbox{max}(0, A_{i+\frac{1}{2}}) - \mbox{min}(0,
          A_{i-\frac{1}{2}}), \Deqlab{eq: 29} \\
          Q_{i}^{-} &=& (\rho _{i}-\rho _{i}^{min})\Delta x, \Deqlab{eq: 30} \\
          R_{i}^{-} &=& \left\{
          \begin{array}{lcl}
            \mbox{min}(1, Q_{i}^{-}/P_{i}^{-}) & \mbox{if} & P_{i}^{-}>0, \\
            0 & \mbox{if} & P_{i}^{-}=0. 
          \end{array}
          \right. \Deqlab{eq: 31} 
        \end{eqnarray}

        $R_{i}^{-}$ $B$O(B $i$ $BE@$+$iN.=P$7$&$k(B antidifuusive $B%U%i%C%/%9$N>e(B
        $B8B$r7h$a$k(B. 

      \item $B>e$GM?$($?(B $R_{i}^{+}, R_{i}^{-}$ $B$rMQ$$$FJd@578?t$r0J2<$N$h(B
        $B$&$KM?$($k(B.
        \begin{equation}
          C_{i+\frac{1}{2}} = \left\{
            \begin{array}{lcl}
               \mbox{min}(R_{i+1}^{+}, R_{i}^{-}) & \mbox{if} &
              A_{i+\frac{1}{2}}\ge 0, \\
               \mbox{min}(R_{i}^{+}, R_{i+1}^{-}) & \mbox{if} &
              A_{i+\frac{1}{2}} < 0.
            \end{array}
            \right. \Deqlab{eq: 32}
        \end{equation}
    \end{enumerate}



