0914 schemes’ consistency and stability - 편집 역사

Admin: /* ⑤ Lax-Friedrich */

2017-09-14T08:35:34Z

⑤ Lax-Friedrich

2017-09-14T07:42:47Z

⑤ Lax-Friedrich

2017-09-13T17:24:26Z

⑥ Crank-Nicolson

2017-09-13T17:23:14Z

⑥ Crank-Nicolson

2017-09-13T17:21:46Z

⑥ Crank-Nicolson

2017-09-13T17:19:49Z

⑥ Crank-Nicolson

2017-09-13T17:11:14Z

⑥ Crank-Nicolson

2017-09-13T17:08:53Z

⑥ Crank-Nicolson

2017-09-13T17:05:17Z

⑥ Crank-Nicholson

2017-09-13T17:04:47Z

@@ 74번째 줄: / 74번째 줄: @@
 ==⑤ Lax-Friedrich==
 $$
-P_{h,k} ϕ= \color{indianred}{ \frac{ϕ^+ - \frac12(ϕ_+-ϕ_-)}{k}} + a\frac{ϕ_+ -ϕ_-}{2h}  \tag8
+P_{h,k} ϕ= \color{indianred}{ \frac{ϕ^+ - \frac12(ϕ_+ +ϕ_-)}{k}} + a\frac{ϕ_+ -ϕ_-}{2h}  \tag8
 $$
 About the first term (red part) ,
 $$\begin{align*} ϕ^+ &= ϕ + kϕ_t + \frac{k^2}2ϕ_{tt} + O(k^3)  \\
-\frac12(ϕ_+-ϕ_-) &= ϕ+ \frac{h^2}2ϕ_{xx} + O(h^3)  \end{align*} $$
+\frac12(ϕ_+ +ϕ_-) &= ϕ+ \frac{h^2}2ϕ_{xx} + O(h^3)  \end{align*} $$
 so the first term is,
 $$  \frac1k \left\{ kϕ_t  + \frac{k^2}2ϕ_{tt} + O(k^3) - \frac{h^2}2ϕ_{xx} + O(h^3)\right\} \\

@@ 78번째 줄: / 78번째 줄: @@
 About the first term (red part) ,
 $$\begin{align*} ϕ^+ &= ϕ + kϕ_t + \frac{k^2}2ϕ_{tt} + O(k^3)  \\
-\frac12(ϕ_+-ϕ_-) &= ϕ+ \frac{h^2}2ϕ_{xx} + O(h^3)  \end{align*} $$ so the first term is,
+\frac12(ϕ_+-ϕ_-) &= ϕ+ \frac{h^2}2ϕ_{xx} + O(h^3)  \end{align*} $$
 $$  \frac1k \left\{ kϕ_t  + \frac{k^2}2ϕ_{tt} + O(k^3) - \frac{h^2}2ϕ_{xx} + O(h^3)\right\} \\
-=ϕ_t + \frac k2 ϕ_{tt} + O(k^2) - \frac{h^2}{2k}ϕ_{xx} + \frac{h^2}k O(h^2) $$
+=ϕ_t + \frac k2 ϕ_{tt} + O(k^2) - \frac{h^2}{2k}ϕ_{xx} + \frac{h^2}k O(h) $$
 Finally, eq.(8) is
 $$

@@ 101번째 줄: / 101번째 줄: @@
 By (9), (10) :
 $$ \frac{v_+^+ - v_-^+ + v_+ - v_-}{4h} = \frac{(9)+(10)}{4h} = \color{royalblue}{\frac1{4h}\left(4hϕ_x + 2khϕ_{xt} + k^2hϕ_{ttx} + O(k^3h^3)\right)} $$
-So,
+So, (blue parts are same)
 $$ P_{k,h}ϕ = ϕ_t + \frac k2 ϕ_{tt} + O(k^2) + a\left( \color{royalblue}{ ϕ_x + \frac k2 ϕ_{xt} + \frac{k^2}4 ϕ_{ttx} + O(k^3h^2) }  \right) $$
 Consistent.
 <disqus/>

@@ 103번째 줄: / 103번째 줄: @@
 So,
 $$ P_{k,h}ϕ = ϕ_t + \frac k2 ϕ_{tt} + O(k^2) + a\left( \color{royalblue}{ ϕ_x + \frac k2 ϕ_{xt} + \frac{k^2}4 ϕ_{ttx} + O(k^3h^2) }  \right) $$
 <disqus/>

@@ 100번째 줄: / 100번째 줄: @@
 $$v_+ - v_- = 2hϕ_x + O(h^3)\tag{10}$$
 By (9), (10) :
-$$ \frac{v_+^+ - v_-^+ + v_+ - v_-}{4h} = \frac{(9)+(10)}{4h} = \color{indianblue}{\frac1{4h}\left(4hϕ_x + 2khϕ_{xt} + k^2hϕ_{ttx} + O(k^3h^3)\right)} $$
+$$ \frac{v_+^+ - v_-^+ + v_+ - v_-}{4h} = \frac{(9)+(10)}{4h} = \color{royalblue}{\frac1{4h}\left(4hϕ_x + 2khϕ_{xt} + k^2hϕ_{ttx} + O(k^3h^3)\right)} $$
 So,
-$$ P_{k,h}ϕ = ϕ_t + \frac k2 ϕ_{tt} + O(k^2) + a\left( \color{indianblue}{ ϕ_x + \frac k2 ϕ_{xt} + \frac{k^2}4 ϕ_{ttx} + O(k^3h^2) }  \right) $$
+$$ P_{k,h}ϕ = ϕ_t + \frac k2 ϕ_{tt} + O(k^2) + a\left( \color{royalblue}{ ϕ_x + \frac k2 ϕ_{xt} + \frac{k^2}4 ϕ_{ttx} + O(k^3h^2) }  \right) $$
 <disqus/>

@@ 96번째 줄: / 96번째 줄: @@
 $$ v_+^+ - v_-^+ = \ 2hϕ_x + 2khϕ_{xt} + k^2h ϕ_{ttx} + O(k^3h^3) \tag9$$
 and
-$$ v_\pm = ϕ\pm hϕ_x + \frac{h^2}2ϕ_{xx} + O(h^3) \tag10$$
+$$ v_\pm = ϕ\pm hϕ_x + \frac{h^2}2ϕ_{xx} + O(h^3) $$
 <disqus/>

@@ 93번째 줄: / 93번째 줄: @@
 P_{k,h}ϕ = \frac{v^+ - v}k + a \frac{v^+_+ - v^+_- + v_+ - v_-}{4h} = 0
 $$
+Refer (4.1)
+$$ v_+^+ - v_-^+ = \ 2hϕ_x + 2khϕ_{xt} + k^2h ϕ_{ttx} + O(k^3h^3) \tag9$$
 <disqus/>

@@ 89번째 줄: / 89번째 줄: @@
 This scheme is consistent only if \(\displaystyle\frac{h^2}k\) goes to \(0\) when \(h, k \to 0\).
-==⑥ Crank-Nicholson ==
+==⑥ Crank-Nicolson ==
 <disqus/>

← 이전 판		2017년 9월 13일 (수) 17:04 판
70번째 줄:		70번째 줄:
	&= ϕ_t + \frac{k^2}6 ϕ_{ttt} + aϕ_x + a\frac{h^2}6 ϕ_{xxx} + O(k^3) + aO(h^3)		&= ϕ_t + \frac{k^2}6 ϕ_{ttt} + aϕ_x + a\frac{h^2}6 ϕ_{xxx} + O(k^3) + aO(h^3)
	\end{align*}$$		\end{align*}$$
		+	Consistent.
		+
		+	==⑤ Lax-Friedrich==
		+	$$
		+	P_{h,k} ϕ= \color{indianred}{ \frac{ϕ^+ - \frac12(ϕ_+-ϕ_-)}{k}} + a\frac{ϕ_+ -ϕ_-}{2h} \tag8
		+	$$
		+	About the first term (red part) ,
		+	$$\begin{align*} ϕ^+ &= ϕ + kϕ_t + \frac{k^2}2ϕ_{tt} + O(k^3) \\
		+	\frac12(ϕ_+-ϕ_-) &= ϕ+ \frac{h^2}2ϕ_{xx} + O(h^3) \end{align*} $$ so the first term is,
		+	$$ \frac1k \left\{ kϕ_t + \frac{k^2}2ϕ_{tt} + O(k^3) - \frac{h^2}2ϕ_{xx} + O(h^3)\right\} \\
		+	=ϕ_t + \frac k2 ϕ_{tt} + O(k^2) - \frac{h^2}{2k}ϕ_{xx} + \frac{h^2}k O(h^2) $$
		+	Finally, eq.(8) is
		+	$$
		+	ϕ_t + \frac k2 ϕ_{tt} - \frac{h^2}{2k}ϕ_{xx} + aϕ_x + a\frac{h^2}6 ϕ_{xxx} + aO(h^3) + O(k^2) + \frac{h^2}k O(h)
		+	$$
		+	Notice the \(\displaystyle\frac{h^2}k\) terms.
		+
		+	This scheme is consistent only if \(\displaystyle\frac{h^2}k\) goes to \(0\) when \(h, k \to 0\).
		+
		+	==⑥ Crank-Nicholson ==
		+
		+	<disqus/>