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\Dauthor{���� �O��}
\Ddate{2011 �N 03 �� 17 ��}% ���t�̎w��
\Dtitle{GFD���[�N�m�[�g}      % ����w�b�_�[�w��
\Dfile{2011-03-ogihara.tex}
\markright{��10�� 1�����ڗ��������̐��l��@} % �E��w�b�_�̎w��

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\begin{document}
\chapter*{�t�^3:Adams-Bashforce����}
�ȉ��̂悤��1�����ڗ���������, 2�ʂ�Adams-Bashforce�X�L�[����3�ʂ�Adams-Bashforce�����ŉ���.
\begin{align}
	\DP{U(x,t)}{t} + c\DP{U(x,t)}{t} = 0
	\Deqlab{1}
\end{align}
\begin{align*}
	\begin{array}{cl}
		��������:&u(x, 0) = \cos{2\pi x}(0<x<1),\\
		         &u(x,0) = 0.0(1.0<x)\\
		���E����:&u(0, t) = u(1.0, t).
	\end{array}
\end{align*}
������, �����$\displaystyle{c=1.0}$�Ƃ�, ��ԍ����ɂ͓񎟐��x�̒��S�������Ƃ�. 

\section*{2�ʂ�Adams-Bashforce����}
Adams-Bashforce�����͑��i�K�̌����ł���. ���̂��ߑO�̃X�e�b�v��1�i�K�̌�����p���ĉ����K�v������.
�����1�X�e�b�v�ڂ̌v�Z�ł�2�i2�ʌ����ł���z�C���@��p���ĉ���. 
�z�C���@��\Deqref{1}�ɓ��Ă͂߂��,
\begin{align*}
	k_1(i\Delta x, n\Delta t) &= -c\Delta t \frac{U^{n}_{i+1} - U^{n}_{i-1}}{2\Delta x}\\
	k_2(i\Delta x, n\Delta t) &= -c\Delta t \frac{U^{*}_{i+1} - U^{*}_{i-1}}{2\Delta x}\\
	U^{n+1}_{i} &= U_i^{n} + \frac{1}{2}(k_1 + k_2)
\end{align*}
�ƂȂ�. ���̂Ƃ�, 
\begin{align*}
	U^{*}_{i+1} &= U^{n}_{i+1} + k_1((i+1)\Delta x)\\
	U^{*}_{i-1} &= U^{n}_{i-1} + k_1((i-1)\Delta x)
\end{align*}
�ł���. 2�X�e�b�v�ڈȍ~�͂���$U^{n+1}$��2�ʂ�Adams-Bashforce �����ɓ��Ă͂߂邱�Ƃŋ��߂����.
���ۂ�$n+2$�ȍ~��Adams-Bashforce �����ɓ��Ă͂߂��,  
% 2�ʂ�Adams-Bashforce������
% \begin{align*}
% 	U_{n + 1} = x_n + h\left(\frac{3}{2}f_n - \frac{1}{2}f_{n - 1}\right).
% \end{align*}
\begin{align*}
	U^{n+2}_{i} &= U^{n+1}_{i} + \Delta t\left(\frac{3}{2}\cdot \frac{U^{n+1}_{i+1} - U^{n+1}_{i-1}}{2\Delta x}-\frac{1}{2}\cdot \frac{U^{n}_{i+1} - U^{n}_{i-1}}{2\Delta x}\right)
\end{align*}
�ƂȂ�. ���ۂɌv�Z����Ƃ���, 1 �X�e�b�v�ڂŌv�Z����$k_1$��p���邱�ƂŌv�Z�ʂ����炷���Ƃ��ł���. 


\begin{figure}[H]
	\begin{tabular}{cc}
	\begin{minipage}{0.5\hsize}
		%\centering
		\begin{center}
		\includegraphics[scale=0.5,keepaspectratio,clip]{adams0.ps}  
		\caption{$\displaystyle{N=50,\Delta t=\frac{1}{100}t=0}$ }
		\label{1}
		\end{center}
	\end{minipage}
	\begin{minipage}{0.5\hsize}
	\begin{center}
	\includegraphics[scale=0.5,keepaspectratio,clip]{adams5.ps}  
	\caption{$\displaystyle{N=50,\Delta t=\frac{1}{100} t=0.5}$}
	\label{2}
	\end{center}
	\end{minipage}
	\end{tabular}
\end{figure}
�i�q�_��$50$��, ���Ԋi�q�Ԋu��$0.01$, ��ԃX�e�b�v��$0.02$, �Ƃ��Čv�Z����. ���̂Ƃ�Curant ����$0.51$�ł���.
���̌��ʂ��}\ref{1}�Ɛ}\ref{2}�ł���. �}\ref{1}��0�b��, �}\ref{2}��0.5�b���̐}�ł���. 

�ڗ����Ă���̂͌����邪���l���U����������. 

\section*{3�ʂ�Adams-Bashforce����}
���l��3�ʂ�Adams-Bashforce�����̎������߂�. 
1�X�e�b�v�ڂƂQ�X�e�b�v�ڂ̌v�Z�ł�3�i3�ʌ����ł���z�C����3��������p���ĉ���. 
�z�C����3��������\Deqref{1}�ɓ��Ă͂߂��,
\begin{align*}
	k_1(i\Delta x, n\Delta t) &= -c\Delta t \frac{U^{n}_{i+1} - U^{n}_{i-1}}{2\Delta x}\\
	k_2(i\Delta x, n\Delta t) &= -c\Delta t \frac{U^{*}_{i+1} - U^{*}_{i-1}}{2\Delta x}\\
	k_3(i\Delta x, n\Delta t) &= -c\Delta t \frac{U^{**}_{i+1} - U^{**}_{i-1}}{2\Delta x}\\
	U^{n+1}_{i} &= U_i^{n} + \frac{1}{4}(k_1 + \frac{1}{3}k_3)
\end{align*}
�ƂȂ�. ���̂Ƃ�, 
\begin{align*}
	U^{*}_{i+1} &= U^{n}_{i+1} + \frac{1}{3}k_1((i+1)\Delta x)\\
	U^{*}_{i-1} &= U^{n}_{i-1} + \frac{1}{3}k_1((i-1)\Delta x)\\
	U^{**}_{i+1} &= U^{n}_{i+1} + \frac{2}{3}k_2((i+1)\Delta x)\\
	U^{**}_{i-1} &= U^{n}_{i-1} + \frac{2}{3}k_2((i-1)\Delta x)
\end{align*}
�ł���. ���̌v�Z���Q�X�e�b�v�s�����Ƃ�$U^{n+1},U^{n+2}$�𓾂邱�Ƃ��ł���. 3�X�e�b�v�ڈȍ~�͂���$U^{n+1}$��$U^{n+2}$��3�ʂ�Adams-Bashforce �����ɓ��Ă͂߂邱�Ƃŋ��߂����.
���ۂ�$n+3$�ȍ~��Adams-Bashforce �����ɓ��Ă͂߂��,  
% 2�ʂ�Adams-Bashforce������
% \begin{align*}
% 	U_{n + 1} = x_n + h\left(\frac{3}{2}f_n - \frac{1}{2}f_{n - 1}\right).
% \end{align*}
\begin{align*}
	U^{n+3}_{i} &= U^{n+2}_{i} + \frac{\Delta t}{12}\left(23\frac{U^{n+2}_{i+1} - U^{n+2}_{i-1}}{2\Delta x}
	-16\frac{U^{n+1}_{i+1} - U^{n+1}_{i-1}}{2\Delta x}+5\frac{U^{n}_{i+1} - U^{n}_{i-1}}{2\Delta x}\right)
\end{align*}
�ƂȂ�. ���ۂɌv�Z����Ƃ���, 1 �X�e�b�v�ڂƂQ�X�e�b�v��
�Ōv�Z����$k_1$��p���邱�ƂŌv�Z�ʂ����炷���Ƃ��ł���. 


\begin{figure}[H]
	\begin{tabular}{cc}
	\begin{minipage}{0.5\hsize}
		%\centering
		\begin{center}
		\includegraphics[scale=0.5,keepaspectratio,clip]{3adams0.ps}  
		\caption{$\displaystyle{N=50,\Delta t=\frac{1}{100}t=0}$ }
		\label{3}
		\end{center}
	\end{minipage}
		\begin{minipage}{0.5\hsize}
		\begin{center}
		\includegraphics[scale=0.5,keepaspectratio,clip]{3adams5.ps}  
		\caption{$\displaystyle{N=50,\Delta t=\frac{1}{100} t=0.5}$}
		\label{4}
		\end{center}
	\end{minipage}
	\end{tabular}
\end{figure}
������2�ʌ����̎��Ɠ��l�ɂ��Čv�Z�������ʂ��}\ref{3}�Ɛ}\ref{4}�ł���. �}\ref{3}��0�b��, �}\ref{4}��0.5�b���̐}�ł���. 

2�ʌ����̂Ƃ��������l�I���U���}�����Ă���.  
	
	
	
\end{document}