%  
%  $BMzNr(B  2004/08/14  $B?y;39L0lO/(B: $B:n@.3+;O(B
%        2004/09/03  $B>.9b@5;L(B, $BKL<iB@0l(B
%

\chapter{$B6u4VJ}8~$NN%;62=(B}

$B$3$N@a$G$O6u4VHyJ,$NN%;62=$NJ}K!$H$=$N$?$a$KI,MW$H$J$kJ?6QA`:n(B, $B6-3&>r7o(B
$B$NM?$(J}$K$D$$$F@bL@$9$k(B. $BN%;62=$O(B 2 $B<!@:EY:9J,$rMQ$$$F9T$&(B.

\Dfigref{koushi} $B$N6u4V$N0LCV$rI=$9E:;z$H$7$F(B, 
$x$ $BJ}8~%U%i%C%/%9$N3J;RE@$r(B ($i(u),k$), 
$z$ $BJ}8~%U%i%C%/%9$N3J;RE@$r(B ($i,k(w)$), 
$B%9%+%i!<NL$N3J;RE@$r(B ($i,k$), 
$B3J;R$N3Q$KEv$?$kE@$r(B ($i(u), k(w)$) $B$H$9$k(B
(\Dfigref{koushi} $B;2>H(B). 
$BC"$7(B $1 \leq i(u) \leq im$, 
$1 \leq i \leq im$, 
$1 \leq k(w) \leq km$, 
$1 \leq k \leq km$ $B$G$"$k(B. 


\section{$BJ?6QA`:n(B}

$B6u4VHyJ,$NN%;62=$r9T$&A0$K(B, $B$=$N$?$a$KI,MW$H$J$kJ?6QA`:n$rDj5A$7$F$*$/(B.
$BNc$($P(B $x$ $BJ}8~%U%i%C%/%93J;RE@$GI>2A$5$l$kJQ?t$r%9%+%i!<NL$N3J;RE@$GI>(B
$B2A$9$k>l9g$O(B, $B%U%i%C%/%93J;RE@$NCM$rJ?6Q$7$F%9%+%i!<3J;RE@$G$NCM$H$_$J$9(B.

$BI,MW$H$J$kJ?6QA`:n$r0J2<$K<($9(B. $B$3$3$G$O(B $x$ $BJ}8~$N%U%i%C%/%93J;RE@$NJQ(B
$B?t$r(B $u_{i(u), k}$, $z$ $BJ}8~$N%U%i%C%/%93J;RE@$NJQ?t$r(B $w_{i, k(w)}$, $B%9(B
$B%+%i!<3J;RE@$NJQ?t$r(B $\pi_{i, k}$ $B$H$7$F$$$k(B.
%
\begin{eqnarray}
&&\pi_{i(u),k} \equiv \frac{\pi_{i+1,  k} + \pi_{i, k}}{2} \\
&&\pi_{i,k(w)} \equiv \frac{\pi_{i, k+1} + \pi_{i, k}}{2}\\
&&\pi_{i(u),k(w)} \equiv 
 \frac{\pi_{i, k} + \pi_{i+1, k}+ \pi_{i, k+1}+ \pi_{i+1, k+1}}{4}\\
&&u_{i,k} \equiv \frac{u_{i(u),  k} + u_{i-1(u), k}}{2} \\
&&u_{i,k(w)} \equiv \frac{u_{i(u), k+1} + u_{i-1(u), k+1} 
 + u_{i(u), k} + u_{i-1(u), k}}{4} \\
&&u_{i(u),k(w)} \equiv \frac{u_{i(u), k+1} + u_{i(u), k}}{2} \\ 
&&w_{i,k} \equiv \frac{w_{i,  k(w)} + w_{i, k-1(w)}}{2} \\
&&w_{i(u),k} \equiv \frac{w_{i+1, k(w)} + w_{i, k(w)} 
 + w_{i+1, k-1(w)} + w_{i, k-1(w)}}{4} \\
&&w_{i(u),k(w)} \equiv \frac{w_{i+1, k(w)} + w_{i, k(w)}}{2} 
\end{eqnarray}
%


\section{$B6u4VHyJ,$NN%;62=(B}

$BG$0U$NJQ?t(B $\phi$ $B$N6u4VHyJ,$r(B 2 $B<!@:EY:9J,$GN%;62=$9$k$H0J2<$N$h$&$K$J(B
$B$k(B.
%
\begin{eqnarray}
&& \left[\DP{\phi}{x} \right]_{i,k} 
 \equiv \frac{\phi_{i(u), k} - \phi_{i-1(u), k}}{\Delta x} \\
&& \left[\DP{\phi}{z} \right]_{i,k} 
 \equiv \frac{\phi_{i, k(w)} - \phi_{i, k-1(w)}}{\Delta z} \\
&& \left[\DP{\phi}{x} \right]_{i(u),k} 
 \equiv \frac{\phi_{i+1, k} - \phi_{i, k}}{\Delta x} \\
&& \left[\DP{\phi}{z} \right]_{i(u),k} 
 \equiv \frac{\phi_{i(u), k(w)} - \phi_{i(u), k-1(w)}}{\Delta z} \\
&& \left[\DP{\phi}{x} \right]_{i,k(w)} 
 \equiv \frac{\phi_{i(u), k(w)} - \phi_{i-1(u), k(w)}}{\Delta x} \\
&& \left[\DP{\phi}{z} \right]_{i,k(w)} 
 \equiv \frac{\phi_{i, k+1} - \phi_{i, k}}{\Delta z} 
\end{eqnarray}

$B0J>e$NDj5A$rF'$^$($F3F3J;RE@>e$K$*$1$k%9%+%i!<NL$H%U%i%C%/%9NL$N6u4VHyJ,(B
$B$rN%;62=$9$k$H0J2<$N$h$&$K$J$k(B.

\subsubsection{$B%9%+%i!<3J;RE@>e$G$N6u4VHyJ,(B}
%
\begin{eqnarray}
\left[\DP{\pi}{x} \right]_{i,k}
 &=& \Dinv{2 \Delta x} \left(\pi_{i+1,k} - \pi_{i-1,k} \right) \\
\left[\DP{\pi}{z} \right]_{i,k}
 &=& \Dinv{2 \Delta z} \left(\pi_{i,k+1} - \pi_{i,k-1} \right) \\
\left[\DP{u}{x}  \right]_{i,k}
 &=& \Dinv{\Delta x} \left(u_{i(u),k} - u_{i-1(u),k} \right) \\
\left[\DP{u}{z} \right]_{i,k}
 &=& \Dinv{4 \Delta z} 
 \left(
  u_{i(u), k+1} + u_{i-1(u),k+1} - u_{i(u),k-1} - u_{i-1(u),k-1} 
 \right) \\
 \left[\DP{w}{x} \right]_{i,k}
 &=& \Dinv{4 \Delta x} 
\left(w_{i+1, k(w)} + w_{i+1,k-1(w)} - w_{i-1,k(w)} - w_{i-1,k-1(w)} \right) \\
\left[\DP{w}{z}  \right]_{i,k}
 &=& \Dinv{\Delta z} \left(w_{i, k(w)} - w_{i,k-1(w)} \right) \\
\left[\DP{\pi u}{z}  \right]_{i(u),k}
 &=& \Dinv{\Delta z} \left(\pi _{i(u),k}u_{i(u),k}-\pi_{i-1(u),k}u_{i-1(u),k}
                     \right)  \\
\left[\DP{\pi w}{z}  \right]_{i(u),k}
 &=& \Dinv{\Delta z} \left(\pi _{i,k(w)}w_{i,k(w)}-\pi_{i,k-1(w)}w_{i,k-1(w)}
                     \right) 
\end{eqnarray}

\subsubsection{$x$ $BJ}8~%U%i%C%/%93J;RE@>e$G$N6u4VHyJ,(B}
%
\begin{eqnarray}
\left[\DP{\pi}{x} \right]_{i(u), k}
 &=& \Dinv{\Delta x} \left(\pi_{i+1,k} - \pi_{i,k} \right) \\
\left[\DP{\pi}{z} \right]_{i(u), k}
 &=& \Dinv{4 \Delta z} 
 \left( \pi_{i+1,k+1} + \pi_{i,k+1} - \pi_{i+1,k-1} - \pi_{i,k-1} \right) \\
\left[\DP{u}{x}  \right]_{i(u), k}
 &=& \Dinv{2 \Delta x} \left(u_{i+1(u),k} - u_{i-1(u),k} \right) \\
\left[\DP{u}{z} \right]_{i(u),k}
  &=& \Dinv{2 \Delta z} \left(u_{i(u),k+1} - u_{i(u),k-1} \right) \\
\left[\DP{w}{x} \right]_{i(u),k}
 &=& \Dinv{2 \Delta x} \left(w_{i+1, k(w)} + w_{i+1,k-1(w)} - w_{i,k(w)} - w_{i,k-1(w)} \right) \\
\left[\DP{w}{z}  \right]_{i(u),k}
 &=& \Dinv{2 \Delta z} \left(w_{i+1, k(w)} + w_{i,k(w)} - w_{i+1,k-1(w)} - w_{i,k-1(w)} \right) \\
\left[\DP[2]{u}{x}  \right]_{i(u),k}
 &=& \Dinv{{\Delta x}^2} \left(u_{i+1(u),k} - 2 u_{i(u),k} + u_{i-1(u),k} \right) \\
\left[\DP[2]{u}{z}  \right]_{i(u),k}
 &=& \Dinv{{\Delta z}^2} \left(u_{i(u),k+1} - 2 u_{i(u),k} + u_{i(u),k-1} \right) 
\end{eqnarray}
%


\subsubsection{$z$ $BJ}8~%U%i%C%/%93J;RE@>e$G$N6u4VHyJ,(B}
%
\begin{eqnarray}
\left[\DP{\pi}{x}\right]_{i,k(w)}
 &=& \Dinv{4 \Delta x} \left( \pi_{i+1,k} + \pi_{i+1,k+1} - \pi_{i-1,k} - \pi_{i-1,k+1} \right) \\
\left[\DP{\pi}{z} \right]_{i,k(w)}
 &=& \Dinv{\Delta z} \left(\pi_{i,k+1} - \pi_{i,k} \right) \\
\left[\DP{u}{x}  \right]_{i,k(w)}
  &=& \Dinv{2 \Delta x} \left(u_{i(u),k+1} + u_{i(u),k} 
	 - u_{i-1(u),k+1(w)} - u_{i-1(u),k} \right) \\
\left[\DP{u}{z} \right]_{i,k(w)}
  &=& \Dinv{2 \Delta z} 
  \left(u_{i(u),k+1} + u_{i-1(u),k+1} - u_{i(u),k} - u_{i-1(u),k} \right) \\
\left[\DP{w}{x} \right]_{i,k(w)}
 &=& \Dinv{2 \Delta x} \left(w_{i+1, k(w)} - w_{i-1,k(w)} \right) \\
 \left[\DP{w}{z}  \right]_{i,k(w)}
 &=& \Dinv{2 \Delta x} \left(w_{i,k+1(w)} - w_{i,k-1(w)} \right) \\
\left[\DP[2]{w}{x}  \right]_{i,k(w)}
 &=& \Dinv{{\Delta x}^2} \left(w_{i+1,k(w)} - 2 w_{i,k(w)} + w_{i-1,k(w)} \right) \\
\left[\DP[2]{w}{z}  \right]_{i,k(w)}
 &=& \Dinv{{\Delta z}^2} 
 \left(w_{i,k+1(w)} - 2 w_{i,k(w)} + w_{i,k-1(w)}
 \right) 
\end{eqnarray}
%



