%
% $BI=Bj(B   SPMODEL $B%5%s%W%k%W%m%0%i%`%I%-%e%a%s%H(B
%        1 $B<!85(B KdV $BJ}Dx<0%b%G%k(B
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% $BMzNr(B   2004/02/25 $B>.9b(B $B@5;L(B  
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\documentclass[a4j,12pt]{jarticle}
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% $B%W%j%"%s%V%k(B
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\usepackage{Dennou6}                      % dennou-sty-6 $B$r;HMQ(B

\Dtitle[1 $B<!85(B KdV $BJ}Dx<0%b%G%k(B]          % $B%X%C%@It%?%$%H%k(B, $B%b%G%kL>$HF1$8(B
{ {\large SPMODEL $B%5%s%W%k%W%m%0%i%`(B} \\  % $B$3$3$OJQ99$7$J$$(B 
  1 $B<!85(B KdV $BJ}Dx<0%b%G%k(B \\              % $B%b%G%kL>(B
  {\large kdv1.f90, kdv2.f90}             % $B%W%m%0%i%`L>(B
}
%
\Dauthor[$B>.9b(B $B@5;L(B]{$B>.9b(B $B@5;L(B}            % $BCx<T(B
\Ddate{2004 $BG/(B 2 $B7n(B 25 $BF|(B}                % $BF|IU(B
\Dnoparindent                             % $BCJMn$N;z2<$2$r$7$J$$(B
%
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%
% $BK\J83+;O(B
%
% $B4pK\E*$J>O@a9=@.$O0J2<$NDL$j(B. 
%
%   1. $B$O$8$a$K(B 
%   2. $B;YG[J}Dx<07O(B
%   3. $BN%;62=(B
%   4. $BMxMQ%b%8%e!<%k$H$=$NB>$N@_Dj(B
%   5. $B?tCM<B83(B
%   6. $B;29MJ88%(B
%   -. $B<U<-(B
% 
% $BI,MW$K1~$8$F9=@.$rJQ99$9$k$3$H(B.
%
\begin{document}

\maketitle                                % $B%?%$%H%k$N:n@.(B
\tableofcontents                          % $BL\<!$N:n@.(B
\pagebreak                                

%----------------------------------------------------------------------
\Dparskip
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\newpage                                  % $B2~%Z!<%8(B
\section{$B35MW(B}

SPMODEL $B%5%s%W%k%W%m%0%i%`!X(Bkfv1.f90$B!Y$*$h$S!X(Bkfv2.f90$B!Y$KMQ$$$i$l$F$$(B
$B$k4pACJ}Dx<0$H6-3&>r7o(B, $B$*$h$S(B, $B$3$N%W%m%0%i%`$rMQ$$$??tCM<B83$NJ}K!$K(B
$B$D$$$F2r@b$9$k(B. $B4pACJ}Dx<0$O(B, 1 $B<!85$N(B KdV $BJ}Dx<0$G$"$k(B.$B7W;;$O%9%Z%/%H(B
$B%kK!$rMQ$$$F9T$$(B, $BE83+$O%U!<%j%(5i?t$rMQ$$$F9T$&(B.  $B%9%Z%/%H%kJQ49(B
$B$H5UJQ49$*$h$SHyJ,1i;;$K$O(B, SPMODEL $B%i%$%V%i%j(B(spml)$B$rMQ$$$F$$$k(B. $B;~4V(B
$B@QJ,K!$O!X(Bkfv1.f90$B!Y$G$O(B leapfrog $B%9%-!<%`(B, $B!X(Bkfv2.f90$B!Y$G$O(B 
Adams-Bashforth $B%9%-!<%`$rMQ$$$k(B. $B?tCM<B83$G$O?6I}$N0[$J$k(B 2 $B$D$N8IN)(B
$BGH2r$r=i4|CM$KM?$((B, $B$=$l$i$N;~4VH/E8$r7W;;$9$k(B.

{\bf $B%W%m%0%i%`L>(B} \\                  % $B%W%m%0%i%`L>(B
{\footnotesize
kdv1.f90, kdv2.f90
}

{\bf $B%W%m%0%i%`<hF@85(B}\\               % $B%W%m%0%i%`<hF@@h(B
{\footnotesize
http:\slash \slash www.gfd-dennou.org\slash arch\slash spmodel\slash 1d-cyclic-e\slash kdv\slash sample\slash SIGEN.htm
}

{\bf SPMODEL $B%5%s%W%k%W%m%0%i%`L\<!(B}\\ % $B%5%s%W%k%W%m%0%i%`L\<!(B($BJQ99$7$J$$(B)
{\footnotesize
 http:\slash \slash www.gfd-dennou.org\slash arch\slash spmodel\slash sample.htm 
}

{\bf SPMODEL $B$N;H$$J}(B}\\               % SPMODEL $B$N;H$$J}(B($BJQ99$7$J$$(B)
{\footnotesize
http:\slash \slash www.gfd-dennou.org\slash arch\slash spmodel\/
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage                                  % $B2~%Z!<%8(B
\section{$B;YG[J}Dx<07O(B}
$B$3$3$G$O;YG[J}Dx<07O$H6-3&>r7o$r5-$9(B. 

\subsection{$B;YG[J}Dx<07O(B}
$B;YG[J}Dx<07O$O(B 1 $B<!85$N(B KdV $BJ}Dx<0$G$"$k(B: 
\begin{eqnarray}
  \DP{\zeta}{t} + \zeta\DP{\zeta}{x} + \DP[3]{\zeta}{x} = 0.
\end{eqnarray}
$B3F5-9f$O0J2<$NNL$r$"$i$o$9(B ($B0J2<$N$h$&$KI=$K$^$H$a$k(B). 
\begin{table}[h]
 \begin{center}
  \begin{tabular}{c l l  } \hline
   $B5-9f(B    &\qquad\qquad&  $BJQ?t(B/$BJ*M}Dj?t(B   \\ \hline 
   $x$      && $B?eJ?0LCV(B        \\
   $t$      && $B;~4V(B            \\ 
   $\zeta$  && $B?6I}(B            \\  \hline 
  \end{tabular} 
 \end{center}
 \caption{$BJQ?t(B, $BJ*M}Dj?t$NDj5A(B}
% \Dtablab{}
\end{table}

\subsection{$B6-3&>r7o(B}

$B?eJ?J}8~$N6-3&>r7o$O<~4|6-3&>r7o$G$"$k(B. 
$B$9$J$o$A(B, $B?eJ?7W;;NN0h$r(B $x_{m}$ $B$H$9$k$H(B,
\[
  \zeta (x+x_{m}) = \zeta (x),
\]
$B$G$"$k(B.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage                                  % $B2~%Z!<%8(B
\section{$BN%;62=(B}
$B$3$N@a$G$OJ}Dx<07O$N6u4VN%;62=$*$h$S;HMQ$7$?;~4V@QJ,K!$K$D$$$F@bL@$7(B, 
$B%W%m%0%i%`Fb$G<B:]$KMQ$$$i$l$F$$$kJ}Dx<0$r5-=R$9$k(B.

\subsection{$B6u4VN%;62=(B}
$B;YG[J}Dx<0$NN%;6I=8=$O0J2<$N$h$&$K$J$k(B.
\begin{eqnarray}
  \DP{\zeta_{i}(t)}{t}
    + \zeta _{i}(t)\DP{\zeta_{i}(t)}{x}
    + \DP[3]{\zeta_{i}(t)}{x} = 0.
\Deqlab{$B6u4VN%;62=$7$?<0(B}
\end{eqnarray}
$B$3$3$G2<IU$-E:;z(B $i$ $B$O?eJ?3J;RE@(B $x_{i}$ $B>e$G$NCM$r<($9(B. 

\subsection{$B?eJ?J}8~$N%9%Z%/%H%kI=8=(B}
$B6u4VN%;62=$7$?;YG[J}Dx<0(B\Deqref{$B6u4VN%;62=$7$?<0(B}$B$r%9%Z%/%H%kK!$rMQ$$(B
$B$FI=8=$9$k(B. $B6u4VJ}8~$N%9%Z%/%H%kE83+$O%U!<%j%(5i?t$rMQ$$$F9T$&(B. $B$?$@$7(B
$BHs@~7A9`$r07$&>l9g$O(B, $B@h$K3J;RE@>e$G$NHs@~7A9`$NCM$r7W;;$7(B, $B$=$NCM$N%9(B
$B%Z%/%H%k$r5a$a$kJ}K!(B($BJQ49K!(B)$B$rMQ$$$k(B. $B0J2<$G$O(B $k$ $B$r(B $x$ $BJ}8~GH?t(B,
$K$ $B$r@ZCGGH?t(B, $I$ $B$r(B $x$ $BJ}8~3J;RE@?t$H$9$k(B.

$\zeta_{i}(t)$ $B$O%9%Z%/%H%k5UJQ49$K$h$C$F0J2<$N$h$&$KE83+$5$l$k(B:
\begin{equation}
  \zeta_{i}(t) = \sum _{k=0}^{K}
  \exp\left(\frac{2\pi i k x_{i}}{x_{m}}\right)\hat \zeta_{k}.
  \Deqlab{$B%9%Z%/%H%k5UJQ49(B}
\end{equation}
$B%9%Z%/%H%k78?t(B $\hat \zeta_{k}$ $B$O0J2<$N%9%Z%/%H%kJQ49$K$h$C$FM?$($i$l$k(B:
\begin{equation}
  \hat \zeta_{k} = \frac{1}{I}
  \sum _{i=0}^{I-1}\exp\left(-\frac{2\pi i k x_{i}}{x_{m}}\right)\zeta_{i}.
  \Deqlab{$B%9%Z%/%H%kJQ49(B}
\end{equation}
\Deqref{$B%9%Z%/%H%k5UJQ49(B}, \Deqref{$B%9%Z%/%H%kJQ49(B}$B$h$j(B, 
\Deqref{$B6u4VN%;62=$7$?<0(B}$B$N%9%Z%/%H%kI=8=$O0J2<$N$h$&$K$J$k(B.
\begin{equation}
  \DP{\hat \zeta_{k}}{t} =
  - \frac{1}{I}
  \sum _{i=0}^{I-1}\exp\left(-\frac{2\pi i x_{i}}{x_{m}}\right)   
  \zeta _{i}\left(\DP{\zeta_{i}}{x}\right)_{i} \\
  -  \left(\frac{2\pi i k}{x_{m}}\right)^{3}\hat \zeta_{k}.
  \Deqlab{$B;YG[J}Dx<0$N%9%Z%/%H%kI=8=(B}
\end{equation}


\subsection{$B;~4V@QJ,(B}
$B$3$3$G$O;~4V@QJ,K!$K$D$$$F5-=R$7(B, $B%W%m%0%i%`Fb$G<B:]$KMQ$$$i$l$F$$$kJ}(B
$BDx<0$r5-=R$9$k(B. $B0J2<$G$O(B $\Delta t$ $B$r;~4V3J;R4V3V(B, $B;~9o(B $\tau \Delta t$ 
$B$K$*$1$k(B $\hat \zeta_{k}$ $B$NCM$r(B $\hat \zeta_{k}^{\tau}$ $BEy$HI=$9(B.

\subsubsection{leapfrog $B%9%-!<%`(B}
leapfrog $B%9%-!<%`$rMQ$$$F(B\Deqref{$B;YG[J}Dx<0$N%9%Z%/%H%kI=8=(B}$B<0$r;~4VJ}(B
$B8~$KN%;62=$9$k$H(B, $B0J2<$N$h$&$KI=$5$l$k(B. 
\begin{equation}
   \hat \zeta_{k}^{\tau +1} = \hat \zeta_{k}^{\tau -1} 
     + 2\Delta t \hat F_{k}^{\tau}.
\end{equation}
$B$3$3$G(B,
\begin{equation}
  \hat F_{k}^{\tau} = 
  - \frac{1}{I}
  \sum _{i=0}^{I-1}\exp\left(-\frac{2\pi i x_{i}}{x_{m}}\right)   
  \zeta _{i}^{\tau}\left(\DP{\zeta_{i}}{x}\right)_{i}^{\tau}
  -  \left(\frac{2\pi i k}{x_{m}}\right)^{3}\hat \zeta_{k}^{\tau}
  \Deqlab{$B;~4V@QJ,1&JU(B}
\end{equation}
$B$G$"$k(B. 

1 $B%9%F%C%WL\$r7W;;$9$k>l9g$O(B Euler $B%9%-!<%`$rMQ$$$k(B.


\subsubsection{Adams-Bashforth $B%9%-!<%`(B}

Adams-Bashforth $B%9%-!<%`$rMQ$$$F(B\Deqref{$B;YG[J}Dx<0$N%9%Z%/%H%kI=8=(B}$B<0(B
$B$r;~4VJ}8~$KN%;62=$9$k$H(B, $B0J2<$N$h$&$KI=$5$l$k(B.
\begin{equation}
  \hat \zeta_{k}^{\tau +1} = \hat \zeta_{k}^{\tau } 
     + \frac{\Delta t}{2} \left( 3\hat F_{k}^{\tau} - 
                       \hat F_{k}^{\tau -1}\right).
\end{equation}
$F_{k}^{\tau}$ $B$NI=8=$O(B\Deqref{$B;~4V@QJ,1&JU(B}$B<0$HF1$8$G$"$k(B.

1 $B%9%F%C%WL\$r7W;;$9$k>l9g$O(B Euler $B%9%-!<%`$rMQ$$$k(B.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage                                  % $B2~%Z!<%8(B
\section{$B;HMQ%b%8%e!<%k$H$=$NB>$N@_Dj(B}

$B%9%Z%/%H%kJQ49$H5UJQ49(B, $BHyJ,1i;;$O(B SPMODEL $B%i%$%V%i%j(B (spml) $B$N(B
ae\_module $B$K4^$^$l$k4X?t$rMQ$$$F9T$&(B. spml $B$,2<0L$G;HMQ$9$k(B ISPACK $B$N(B
$B;EMM$+$i(B, $B3J;RE@?t(B $I$ $B$O6v?t$G(B, $B$+$D(B $I/2=2^{a}3^{b}5^{c}$ ($a, b, c$ 
$B$O(B 0 $B$^$?$O@0?t(B) $B$G$J$1$l$P$J$i$J$$(B. $BHs@~7A9`$N7W;;$K$h$C$F@8$8$k%(%j(B
$B%"%8%s%0$rKI$0$?$a(B, $B3J;RE@?t(B $I$ $B$H@ZCGGH?t(B $K$ $B$O(B $I>3K$ $B$rK~$?$9$h$&(B
$B$KM?$($k(B.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage                                  % $B2~%Z!<%8(B
\section{$B?tCM<B83(B}

$B=i4|$K?6I}$N0[$J$k(B 2 $B$D$N8IN)GH$rM?$((B, $B$=$l$i$N;~4VH/E8$r7W;;$9$k(B. $B=i(B
$B4|>r7o$O0J2<$N$h$&$KM?$($k(B(\Dfigref{init}).
\begin{equation}
  \zeta = U_{1}\mbox{sech}^{2}\left(\frac{x-x_{1}}{\sqrt{12/U_{1}}}\right)
        + U_{2}\mbox{sech}^{2}\left(\frac{x-x_{2}}{\sqrt{12/U_{2}}}\right).
  \Deqlab{$B=i4|>r7o(B}
\end{equation}
$U_{1}=720, U_{2}=1440, x_{1}=x_{m}/2, x_{2}=x_{m}/4$ $B$H$9$k(B. $B?eJ?7W;;(B
$BNN0h$NBg$-$5$O(B $x_{m}=3$ $B$H$9$k(B.

$B3J;RE@?t(B $I$ $B$H@ZCGGH?t(B $K$ $B$O$=$l$>$l(B $I=128, K=42$ $B$H$9$k(B. $B;~4V3J;R(B
$B4V3V$O(B, $B!X(Bkdv1.f90$B!Y(B (leapfrog $B%9%-!<%`$rMQ$$$k>l9g(B)$B$G$O(B $\Delta
t=10^{-6}$ sec, $B!X(Bkdv2.f90$B!Y(B (Adams-Bashforth $B%9%-!<%`$rMQ$$$k>l9g(B)$B$G(B
$B$O(B$\Delta t=2\times 10^{-7}$ sec $B$G$"$k(B. $B7W;;%9%F%C%W?t$O$=$l$>$l(B 
5,000$B$H(B 250,000 $B$G$"$k(B.

\Dfigref{kdv1},\Dfigref{kdv2}$B$K7W;;7k2L$r<($9(B. $B?6I}$,Bg$-$$$[$I0LAjB.(B
$BEY$bBg$-$$$3$H(B, $B?6I}$NBg$-$$GH$,?6I}$N>.$5$$GH$rDI$$1[$7$F9T$/MM;R$,$o(B
$B$+$k(B.

\begin{figure}[b]
\begin{center}
\Depsf[][90mm]{./figs/init.ps}
\caption{$B=i4|CM$NJ,I[(B.}
\Dfiglab{init}
\end{center}
\end{figure}

\begin{figure}[p]
\begin{center}
\Depsf[][100mm]{./figs/kdv1.ps}
\caption{Leapfrog $B%9%-!<%`$rMQ$$$?>l9g$N?tCM7W;;$N7k2L(B. $x$-$t$ $BCGLL?^(B.} 
\Dfiglab{kdv1}

\Depsf[][100mm]{./figs/kdv2.ps}
\caption{Adams-Bashforth + Crank Nicholson $B%9%-!<%`$rMQ$$$?>l9g$N?tCM7W;;$N7k2L(B. $x$-$t$ $BCGLL?^(B.} 
\Dfiglab{kdv2}
\end{center}
\end{figure}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage                                  % $B2~%Z!<%8(B
\newpage
\section{$B;29MJ88%(B}
\begin{description}
 \item  Witham, G. B., 1974: 
        {\it Linear and nonlinear waves}, John Wiley \& Sons, 636pp.

 \item $BC'(B $BM'@5(B, 1982: $BN.BNNO3X(B, $BG]Iw4[(B.

 \item $BC]9-??0l(B, $B@P2,7=0l(B, $BK-ED1Q;J(B, $B@PEO@5<y(B, $BNS>M2p(B, SPMODEL $B3+H/%0%k!<%W(B, 
       2002: 
       $B3,AXE*CO5eN.BNNO3X%9%Z%/%H%k%b%G%k=8(B (SPMODEL),\\
       http:\slash \slash www.gfd-dennou.org\slash arch\slash spmodel\slash ,
       $BCO5eN.BNEEG>6f3ZIt(B. 
\end{description}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage                                  % $B2~%Z!<%8(B
\section*{$B<U<-(B}                           % $B<U<-(B. $B$3$N$^$^MxMQ$9$k(B.
\markright{$B<U<-(B}                          % $B%X%C%@$KI=<((B
\addcontentsline{toc}{section}{$B<U<-(B}      % $BL\<!$KI=<((B

$BK\;q8;$O(B, $BCO5eN.BNEEG>6f3ZIt$N%$%s%?!<%M%C%H>e$G$N3X=QCN<1$N=8@Q$H3hMQ(B
$B$N<B83$N0l4D$H$7$F(B
\begin{center}
 http:\slash \slash www.gfd-dennou.org\slash arch\slash spmodel\slash 
\end{center}
$B$K$*$$$F8x3+$5$l$F$$$k$b$N$G$"$k(B 
(\copyright $BCO5eN.BNEEG>6f3ZIt%9%Z%/%H%k%b%G%k%W%m%8%'%/%H(B 
spmodel@gfd-dennou.org 2002. ). 
$BK\;q8;$O(B, $BCx:n<T$N=t8"Mx$KDq?($7$J$$(B($BLBOG$r$+$1$J$$(B)$B8B$j$K$*$$$F<+M3$K(B
$BMxMQ$7$F$$$?$@$$$F9=$o$J$$(B. $B$J$*(B, $BMxMQ$9$k:]$K$O:#0lEY<+$iFbMF$r3N$+$a(B
$B$k$3$H$r$*4j$$$9$k(B($BL5J]>ZL5@UG$86B'(B).

$BK\;q8;$K4^$^$l$k85;q8;Ds6!<T(B($B?^Ey$NHG85Ey$r4^$`(B)$B$+$i$O(B,  $BD>@\E*$J7A$G(B
$B$N(B WEB $B>e$G$NCx:n8"$^$?$O;HMQ5vBz$rF@$F$$$J$$>l9g$,$"$k$,(B,  $B>!<j$J$,$i(B, 
$B!VL$Mh$N650i!W$N$?$a$N<B83$H$$$&3X=QL\E*$G$"$k$3$H$r$4M}2r$$$?$@$1$k$b(B
$B$N$H?.$8(B,  $B3X=QI8=`$N0zMQ<j=g$r<i$k$3$H$G=t<jB3$-$rN,$5$;$F$$$?$@$$$F(B
$B$$$k(B.  $BK\;q8;$NMxMQ<T$K$O(B, $B$3$NE@$rM}2r$N>e(B,  $BCm0U$7$F07$C$F$$$?$@$1$k(B
$B$h$&$*4j$$$9$k(B.  $BK|0l(B, $BITET9g$N$"$k>l9g$K$O(B
\begin{center}
 spmodel@gfd-dennou.org
\end{center}
$B$^$GO"Mm$7$F$$$?$@$1$l$P9,$$$G$"$k(B. 


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\end{document}

