\section{$B6qBNE*$J7W;;Nc(B}

\subsection{1 $B<!85$N6-3&CM8GDjLdBj(B}

\Deqref{$B3H;6J}Dx<0(B}$B<0$r0J2<$N$h$&$J6-3&>r7o(B, $B=i4|>r7o$rM?$($F2r$/(B. 
$x=0,1$ $B$K$*$1$k6-3&>r7o$O(B,

\begin{equation}
 \psi = 0 \quad ( t \geq 0 ),
\end{equation}

$B$H$9$k(B. $t=0$ $B$K$*$1$k=i4|>r7o$O(B

\begin{eqnarray}
  u=2x, && 0 \leq x \leq \frac{1}{2}, \\
  u=2(1-x), && \frac{1}{2} \leq x \leq 1,
\end{eqnarray}

$B$H$9$k(B. 

$h=0.1, \tau = 0.01 $ $B$H$7$F7W;;$9$k(B. $B$3$N>l9g78?t9TNs$O(B,

\[
\begin{array}{lll}
  A_{ii}    = 2  (h^{2}/\tau +1) = 4, &  A_{i,i-1} = -1,&  A_{i,i+1} = -1, \\
  B_{ii}    = 2 (h^{2}/\tau -1) = 0, & B_{i,i-1} = 1, &  B_{i,i+1} = 1, 
\end{array}
\]

$B$H$J$k(B. $t=0.5$ $B$^$G$N7k2L$r(B $t=0.1$ $B4V3V$G<($7$?$N$,(B\Dfigref{$B?^(B1}$B$G$"$k(B.
$x$ $BJ}8~$K$O(B $x=0.5$ $B$^$G<($7$?(B.

\begin{figure}[hb]
\begin{center}
  \Depsf[120mm][]{./fig1.ps} 
  \caption{1$B<!853H;6J}Dx<0$N7W;;7k2L(B. $t=0.5$ $B$^$G$N7k2L$r(B $t=0.1$ $B4V(B
           $B3V$G<($7$?(B.}
  \Dfiglab{$B?^(B1}
  \end{center} 
\end{figure}

\subsection{$BITEy3J;R4V3V(B, $B3H;678?tHs0lMM$N>l9g(B}

$B6u4VNN0h(B, $B=i4|>r7o(B, $B6-3&>r7o$OA4@a$HF1$8$h$&$KM?$i$l$k$,(B, $B3H;678?t$,6u(B
$B4VE*$KHs0lMM(B, $BITEy3J;R4V3V$N>l9g$K$D$$$F9M$($k(B. $B2r$/$Y$-<0$O(B

\begin{equation}
 \DP{\psi }{t} = \DP{}{x}\left(K\DP{\psi }{x}\right),
\end{equation}

$B$G$"$k(B. 

$BA4@a$HF1MM$K$^$:6u4VJ}8~$KN%;62=$9$k(B. $B$?$@$7(B $\psi, K$ $B$OH>@0?t3J;RE@(B
$B>e$K$H$k(B.

\begin{equation}
 \DP{\psi _{i+\frac{1}{2}}}{t} = \frac{1}{\Delta x_{i+\frac{1}{2}}}
 \left[(K_{i+\frac{3}{2}}+K_{i+\frac{1}{2}})
        \frac{\psi _{i+\frac{3}{2}}-\psi _{i+\frac{1}{2}}}
             {\Delta x_{i+\frac{3}{2}}+\Delta x_{i+\frac{1}{2}}}
      - (K_{i+\frac{1}{2}}+K_{i-\frac{1}{2}})
        \frac{\psi _{i+\frac{1}{2}}-\psi _{i-\frac{1}{2}}}
             {\Delta x_{i+\frac{1}{2}}+\Delta x_{i-\frac{1}{2}}}
 \right].
\end{equation}

$B$@$?$7(B $\Delta x_{i+\frac{1}{2}}=x_{i+1}-x_{i}(i=[0,N-1])$ $B$G$"$k(B.

$B$5$i$K;~4VJ}8~$KN%;62=$9$k(B. 

\begin{eqnarray}
 \frac{\psi _{i+\frac{1}{2}}^{j+1}-\psi _{i+\frac{1}{2}}^{j}}{\Delta t}
 &=&
 \frac{1}{2\Delta x_{i+\frac{1}{2}}}
 \left[(K_{i+\frac{3}{2}}+K_{i+\frac{1}{2}})
        \frac{\psi _{i+\frac{3}{2}}^{j+1}-\psi _{i+\frac{1}{2}}^{j+1}}
             {\Delta x_{i+\frac{3}{2}}+\Delta x_{i+\frac{1}{2}}}
      - (K_{i+\frac{1}{2}}+K_{i-\frac{1}{2}})
        \frac{\psi _{i+\frac{1}{2}}^{j+1}-\psi _{i-\frac{1}{2}}^{j+1}}
             {\Delta x_{i+\frac{1}{2}}+\Delta x_{i-\frac{1}{2}}}
 \right. \nonumber \\
 && \left.
    +  (K_{i+\frac{3}{2}}+K_{i+\frac{1}{2}})
        \frac{\psi _{i+\frac{3}{2}}^{j}-\psi _{i+\frac{1}{2}}^{j}}
             {\Delta x_{i+\frac{3}{2}}+\Delta x_{i+\frac{1}{2}}}
      - (K_{i+\frac{1}{2}}+K_{i-\frac{1}{2}})
        \frac{\psi _{i+\frac{1}{2}}^{j}-\psi _{i-\frac{1}{2}}^{j}}
             {\Delta x_{i+\frac{1}{2}}+\Delta x_{i-\frac{1}{2}}}
 \right].
\end{eqnarray}

$B$3$l$r@0M}$9$k$H(B,

\begin{eqnarray}
&&
  -\frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}
   \frac{\overline{K}_{i}}{\overline{\Delta x}_{i}}
        \psi _{i-\frac{1}{2}}^{j+1}
+ \left[2
  + \frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}\left(
    \frac{\overline{K}_{i+1}}{\overline{\Delta x}_{i+1}}
  +  \frac{\overline{K}_{i}}{\overline{\Delta x}_{i}}\right)\right]
        \psi _{i+\frac{1}{2}}^{j+1}
  -\frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}
   \frac{\overline{K}_{i+1}}{\overline{\Delta x}_{i+1}}
        \psi _{i+\frac{3}{2}}^{j+1} \nonumber \\
&=&
  +\frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}
   \frac{\overline{K}_{i}}{\overline{\Delta x}_{i}}
        \psi _{i-\frac{1}{2}}^{j}
  + \left[2
  - \frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}\left(
    \frac{\overline{K}_{i+1}}{\overline{\Delta x}_{i+1}}
  + \frac{\overline{K}_{i}}{\overline{\Delta x}_{i}}\right)\right]
        \psi _{i+\frac{1}{2}}^{j}
  + \frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}
    \frac{\overline{K}_{i+1}}{\overline{\Delta x}_{i+1}}
        \psi _{i+\frac{3}{2}}^{j},
\end{eqnarray}

$B$H$J$k(B. $B$?$@$7$3$3$G(B 
$\overline{K}_{i}=(K_{i+\frac{1}{2}}+K_{i-\frac{1}{2}})/2,
\overline{\Delta x}_{i}=(\Delta x_{i+\frac{1}{2}}+\Delta x_{i-\frac{1}{2}})/2$
$B$G$"$k(B. $B$3$N<0$b(B\Deqref{$B9TNsI=8=(B}$B<0$N$h$&$K9TNs7A<0$GI=$9$3$H$,$G$-$k(B. 
$B78?t9TNs$NMWAG$O(B,

\begin{eqnarray*}
  && A_{ii} = 2 + \frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}
    \left(
    \frac{\overline{K}_{i+1}}{\overline{\Delta x}_{i+1}}
  + \frac{\overline{K}_{i}}{\overline{\Delta x}_{i}}\right), 
     A_{i,i+1} = -\frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}
   \frac{\overline{K}_{i+1}}{\overline{\Delta x}_{i+1}},
    A_{i,i-1} =   -\frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}
   \frac{\overline{K}_{i}}{\overline{\Delta x}_{i}}, \nonumber \\
 &&  B_{ii} = 2 - \frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}
    \left(
    \frac{\overline{K}_{i+1}}{\overline{\Delta x}_{i+1}}
  + \frac{\overline{K}_{i}}{\overline{\Delta x}_{i}}\right), 
    B_{i,i+1} =  \frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}
   \frac{\overline{K}_{i+1}}{\overline{\Delta x}_{i+1}}, 
    B_{i,i-1} =  \frac{\Delta t}{\Delta x_{i+\frac{1}{2}}}
   \frac{\overline{K}_{i}}{\overline{\Delta x}_{i}},
\end{eqnarray*}

$B$H$J$k(B. $B$?$@$778?t9TNs$NBh(B 1 $B9T5Z$SBh(B N $B9T$NBP3QMWAG$O(B, 

\[
  A_{11} = 2 + \frac{\Delta t}{\Delta x_{\frac{1}{2}}}
    \left(
    \frac{\overline{K}_{1}}{\overline{\Delta x}_{1}}
  + \frac{\overline{K}_{0}}{\Delta x_{\frac{1}{2}}/2}\right), \quad
  B_{11} = 2 - \frac{\Delta t}{\Delta x_{\frac{1}{2}}}
    \left(
    \frac{\overline{K}_{1}}{\overline{\Delta x}_{1}}
  + \frac{\overline{K}_{0}}{\Delta x_{\frac{1}{2}}/2}\right), 
\]
\[
  A_{NN} = 2 + \frac{\Delta t}{\Delta x_{N-\frac{1}{2}}}
    \left(
    \frac{\overline{K}_{N}}{\Delta x_{N-\frac{1}{2}}/2}
  + \frac{\overline{K}_{N-1}}{\overline{\Delta x}_{N-1}}\right), \quad
  B_{NN} = 2 - \frac{\Delta t}{\Delta x_{N-\frac{1}{2}}}
    \left(
    \frac{\overline{K}_{N}}{\Delta x_{N-\frac{1}{2}}/2}
  + \frac{\overline{K}_{N-1}}{\overline{\Delta x}_{N-1}}\right), 
\]

$B$H$J$k(B.

$BA4$F$NHO0O$G(B $K=1.0$ $B$H$7$?>l9g$N7W;;7k2L$r(B\Dfigref{$B?^(B2}$B$K(B, $x>0.5$ $B$G(B 
$K=3.0$ $B$H$7$?>l9g$N7W;;7k2L$r(B\Dfigref {$B?^(B3}$B$K<($9(B.

\begin{figure}[hb]
\begin{center}
  \Depsf[120mm][]{./fig2.ps} 
  \caption{1$B<!853H;6J}Dx<0$N7W;;7k2L(B. $t=2.0$ $B$^$G$N7k2L$r(B $t=0.1$ $B4V(B
           $B3V$G<($7$?(B.}
  \Dfiglab{$B?^(B2}

  \Depsf[120mm][]{./fig3.ps} 
  \caption{1$B<!853H;6J}Dx<0$N7W;;7k2L(B. $x>0$ $B$G(B $K=3.0$ $B$H$7$?>l9g(B.
           $B$=$NB>$O?^(B2$B$HF1$8(B.}
  \Dfiglab{$B?^(B3}
  \end{center} 
\end{figure}


