\documentclass[a4j,12pt]{jreport}
\usepackage{Dennou6}
\usepackage{times}
\usepackage{ascmac}
\usepackage{float}

\Dauthor{���� �O��}
\Ddate{2010 �N 11 �� 25 ��}% ���t�̎w��
\Dtitle{GFD���[�N�m�[�g}      % ����w�b�_�[�w��
\Dfile{2010-1111-ogihara.tex}
\markright{�U���������̃z�C���X�L�[���Ə���X�L�[��} % �E��w�b�_�̎w��

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\setlength{\parskip}{1em}   

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 % chapter �̒��O�ŉ��s���Ȃ��悤�ɐݒ�. 
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\begin{document}

\begin{eqnarray*}
	\DD{U}{t} &=& f\\
	f &=& i\omega U
\end{eqnarray*}
�Ƃ���, �����l1, $\omega = \pi$�Ƃ���.���̂Ƃ��̏���X�L�[���ƃz�C���X�L�[���̐U���������Ă݂�.

\subsection*{����X�L�[��}
�U���������̏���X�L�[����
\begin{eqnarray*}
	U^{(n + 1)^{*}} = U^{(n)} + i\omega \Delta tU^{(n)}.
\end{eqnarray*}
���ɑ����l�����߂�.
\begin{eqnarray*}
	U^{(n + 1)} 
	&=& U^{(n)} + i\omega \Delta t\left(U^{(n + 1)^{*}}\right)\\
	&=& U^{(n)} + i\omega \Delta t\left(U^{(n)} + i\omega \Delta tU^{(n)}\right)\\
	&=& \left(1 + i\omega \Delta t+ (i\omega \Delta t)^2\right)U^{(n)}.
\end{eqnarray*}
������$\displaystyle{\lambda \equiv \frac{U^{(n + 1)}}{U^{(n)}}}$�Ƃ����
\begin{eqnarray*}
	\lambda = 1 + ip - p^2.
\end{eqnarray*}
�A��, $p \equiv \omega \Delta t$.

$\displaystyle{\Delta t = \frac{2}{100}}$��, $\displaystyle{\Delta t = \frac{2}{10}}$�Ƃ��čs�������ʂ�\�ɂ܂Ƃ߂�. 
\begin{center}
	\begin{tabular}{|c|c|c|c|c|}
		\hline
		 & ���l���̑�����&�������̒�`&Re[$U$]\\
		\hline
		$\displaystyle{\Delta t = 0.02}$& 0.99803193520755129  & 0.99803193520755129  & 0.82107776\\
		\hline
		$\displaystyle{\Delta t = 0.2}$&0.87239347178370708 &0.87239347178370708& -4.75382917E-02\\
		\hline
	\end{tabular}
\end{center}
	\begin{tabular}{|c|c|}
		\hline
		&$\omega \Delta t$\\
		\hline
		$\displaystyle{\Delta t = 0.02}$& 6.28318530717958679E-002\\
		\hline
		$\displaystyle{\Delta t = 0.2}$&      0.62831853071795862\\		
		\hline
	\end{tabular}

%\begin{center}
\begin{figure}[H]
	\begin{tabular}{cc}
	\begin{minipage}{0.5\hsize}
		%\centering
		\begin{center}
			\includegraphics[scale=0.5,keepaspectratio,]{mazuno100.ps}  
			\caption{$\displaystyle{\Delta t = \frac{2}{100}}$�̏���X�L�[��}
		\end{center}
		\label{ma100}
	\end{minipage}
	\begin{minipage}{0.5\hsize}
		\begin{center}
			\includegraphics[scale=0.5,keepaspectratio,]{mazuno_sa100.ps}  
			\caption{$\displaystyle{\Delta t = \frac{2}{100}}$�̏���X�L�[���Ɖ�͉��Ƃ̍�}
		\end{center}
		\label{ma10}
	\end{minipage}
	\end{tabular}
\end{figure}
%\end{center}
\begin{figure}[H]
	\begin{tabular}{cc}
	\begin{minipage}{0.5\hsize}
		%\centering
		\begin{center}
			\includegraphics[scale=0.5,keepaspectratio,]{mazuno10.ps}  
			\caption{$\displaystyle{\Delta t = \frac{2}{10}}$�̏���X�L�[��}
		\end{center}
		\label{ma100}
	\end{minipage}
	\begin{minipage}{0.5\hsize}
		\begin{center}
			\includegraphics[scale=0.5,keepaspectratio,]{mazuno_sa10.ps}  
			\caption{$\displaystyle{\Delta t = \frac{2}{10}}$�̏���X�L�[���Ɖ�͉��Ƃ̍�}
		\end{center}
		\label{ma10}
	\end{minipage}
	\end{tabular}
\end{figure}
�\���番����悤��$\displaystyle{\Delta t = \frac{2}{100}}$�̂Ƃ��͐��l���̑������Ɖ�͉��̑������͂قڈ�v��1�ɋ߂��Ȃ�. ���̂��߉�͉�$e^{i\omega t}$�ɂقڈ�v����. ������, 
�����l��������Ă��܂�, ��͉��Ƃ̍����ǂ�ǂ�J���Ă��܂�. �܂�, $\displaystyle{\Delta t = \frac{2}{10}}$�ɉ�����Ɛ��l���̑������Ɖ�͉��̑������͈�v���邪1���痣��Ă��܂�, �ɒl������Ēl���ς���Ă��܂�. ����X�L�[����
�ʑ�����͉����, �����i��. ���ꂪ$\displaystyle{\Delta t = \frac{2}{10}}$���Ɣ@���ɕ\��Ă��邽�߂ł���.  

\subsection*{�z�C���X�L�[��}
�z�C���X�L�[���̑����W���͏���X�L�[���Ɠ��l�ɋ��߂��
\begin{eqnarray*}
	U^{(n + 1)} 
	&=& U^{(n)} + i\omega \Delta t\left(\frac{1}{2} U^{(n)} + \frac{1}{2} U^{(n + 1)^{*}}\right)\\
	&=& U^{(n)} + i\omega \Delta t\left(\frac{1}{2} U^{(n)} + \frac{1}{2} U^{(n)} + i\omega\Delta t\frac{1}{2} U^{(n)}\right)\\
	&=& (1 + ip - \frac{1}{2}p^2 )U^{(n)}.
\end{eqnarray*}
\begin{eqnarray*}
	\lambda = 1 + ip - \frac{1}{2}p^2.
\end{eqnarray*}

���l��$\displaystyle{\Delta t = \frac{2}{100}}$��, $\displaystyle{\Delta t = \frac{2}{10}}$�Ƃ��čs�������ʂ�\�ɂ܂Ƃ߂�.
\begin{center}
	\begin{tabular}{|c|c|c|c|c|}
		\hline
		 & ���l���̑�����&�������̒�`&Re[$U$]\\
		\hline
		$\displaystyle{\Delta t = 0.02}$& 1.0000019481799232  &   1.0000019481799229     & 1.0001863\\
		\hline
		$\displaystyle{\Delta t = 0.2}$&   1.0192956570169427 &1.0192956570169427 & 1.1335322\\
		\hline
	\end{tabular}
\end{center}
	\begin{tabular}{|c|c|}
		\hline
		&$\omega \Delta t$\\
		\hline
		$\displaystyle{\Delta t = 0.02}$& 6.28318530717958679E-002\\
		\hline
		$\displaystyle{\Delta t = 0.2}$& 0.62831853071795862\\		
		\hline
	\end{tabular}

\begin{figure}[H]
	\begin{minipage}{0.5\hsize}
		\centering
		%\begin{center}
			\includegraphics[scale=0.5,keepaspectratio,clip]{heun100.ps}  
			\caption{$\displaystyle{\Delta t = \frac{2}{100}}$�̃z�C���X�L�[��}
		%\end{center}
		\label{he100}
	\end{minipage}
	\begin{minipage}{0.5\hsize}
		%\begin{center}
			\includegraphics[scale=0.5,keepaspectratio,clip]{heun_sa100.ps}  
			\caption{$\displaystyle{\Delta t = \frac{2}{100}}$�̃z�C���X�L�[���Ɖ�͉��Ƃ̍�}
		%\end{center}
		\label{he10}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}{0.5\hsize}
		\centering
		%\begin{center}
			\includegraphics[scale=0.5,keepaspectratio,clip]{heun10.ps}  
			\caption{$\displaystyle{\Delta t = \frac{2}{10}}$�̃z�C���X�L�[��}
		%\end{center}
		\label{he100}
	\end{minipage}
	\begin{minipage}{0.5\hsize}
		%\begin{center}
			\includegraphics[scale=0.5,keepaspectratio,clip]{heun_sa10.ps}  
			\caption{$\displaystyle{\Delta t = \frac{2}{10}}$�̃z�C���X�L�[���Ɖ�͉��Ƃ̍�}
		%\end{center}
		\label{he10}
	\end{minipage}
\end{figure}
�}����$\displaystyle{\Delta t = \frac{2}{100}}$�͓r���ŉ�͉��Ƃ̍����J���Ă��܂����Ō�Ɏ������Ă����قډ�͉��Ɉ�v����. ������, $\displaystyle{\Delta t = \frac{2}{10}}$
�ɂ���ƍŌ�̒l�������l��菭�������Ă��܂�. 
�����, $\left| \lambda \right|$��$\displaystyle{\sqrt{1 + \frac{1}{4}p^4}}$�ł���̂�, ����$p^4$�ɂ���ċN����. 
�܂�, �z�C���X�L�[���͉�͉����ʑ����x���Ȃ�͂���������p����$p$��$p\ll 1$�Ƃ�
�����Ȃ��̂ō����o�Ȃ�.  
%$p=0.6$�Ȃ̂ő����l��$0.6^4$��$10^4$�{�ő����Ă��܂�.

\end{document}