0914 schemes’ consistency and stability
Let, \( v^+ = v_m^{n+1}, v_- = v_{m-1}^n \) and so on.
And usually, \(v = \phi\)
목차
Consistency
‘consistency’ means \(P_{h,k} \phi = P \phi \) as \(h, k \to 0\).
And by Taylor, $$ \begin{align*} ϕ^\pm &= ϕ\pm kϕ_t + \frac{k^2}2 ϕ_{tt} + O(k^3) \tag{1}\\ &= ϕ\pm kϕ_t + \frac{k^2}2 ϕ_{tt} \pm \frac{k^3}6 ϕ_{ttt} + O(k^4) \\
\\
\frac{ϕ^+ - ϕ}k &= ϕ_t + \frac k2ϕ_{tt} + O(k^2) \tag{1.1}\\ \\ ϕ_\pm &= ϕ \pm hϕ_x + \frac{h^2}2 ϕ_{xx} + O(h^3) \tag2\\ &= ϕ \pm hϕ_x + \frac{h^2}2 ϕ_{xx} \pm \frac{h^3}6 ϕ_{xxx} + O(h^4) \\ \\ \frac{ϕ_+ - ϕ}h &= ϕ_x + \frac h2ϕ_{xx} + O(h^2) \tag{2.1 : forward space}\\ \frac{ϕ- ϕ_-}h &= ϕ_x - \frac h2ϕ_{xx} + \frac{h^2}6ϕ_{xxx} + O(h^3) \tag{2.2 : backward space} \\ \frac{ϕ_+ - ϕ_-}{2h} &= ϕ_x + \frac{h^2}6 ϕ_{xxx} + O(h^3) \tag{2.3 : central space} \end{align*}$$
Sometimes we need \(v^+_+\), $$ v_+^+ = (v^+)_+ = v^+ + hv^+_x + \frac{h^2}2 v^+_{xx} + O(h^3) \tag3 $$ Substitue (1) to (3), then, $$\begin{align*} v_+^+ = &\ ϕ + kϕ_t + \frac{k^2}2 ϕ_{tt} +\\
&hϕ_x + khϕ_{xt} + \frac{k^2h}2 ϕ_{ttx} +\\
&\frac{h^2}2ϕ_{xx} + \frac{kh^2}2 ϕ_{txx} + \frac{k^2h^2}4 ϕ_{ttxx} + O(k^3h^3) \tag4 \\
\\ \text{ so, } v_+^+ - v_-^+ =& \ 2hϕ_x + 2khϕ_{xt} + k^2h ϕ_{ttx} + O(k^3h^3) \tag{4.1}
\end{align*}
$$
And let the PDE is \( u_t + au_x = 0 \)
Then, we are ready.
① FTFS
$$\begin{align*} P_{h,k} ϕ &= \frac{ϕ^+ - ϕ}k + a\frac{ϕ_+ -ϕ}h \tag5 \\
&= ϕ_t + \frac k2ϕ_{tt} + aϕ_x + a\frac h2ϕ_{xx} + O(k^2) + aO(h^2) \tag{5.1}
\end{align*}$$ \(Pϕ = ϕ_t + aϕ_x\) , so, $$ P_{h,k}ϕ - Pϕ = \frac k2ϕ_{tt} + a\frac h2ϕ_{xx} + O(k^2) + aO(h^2) \tag6$$ if \(h, k \to 0\), (6) goes to \(0\). So, FTFS scheme is consistent.
② FTBS
$$\begin{align*} P_{h,k} ϕ &= \frac{ϕ^+ - ϕ}k + a\frac{ϕ -ϕ_-}h \tag7 \\
&= ϕ_t + \frac k2ϕ_{tt} + aϕ_x - a\frac h2ϕ_{xx} + O(k^2) + aO(h^2) \tag{7.1}
\end{align*}$$ (7.1) is obtained by substituting (2.2) to (7).
This scheme is consistent. (Think about this : \((7.1) - Pϕ\))
③ FTCS
$$\begin{align*} P_{h,k} ϕ &= \frac{ϕ^+ - ϕ}k + a\frac{ϕ_+ -ϕ_-}{2h} \\
&= ϕ_t + \frac k2ϕ_{tt} + aϕ_x + a\frac{h^2}6 ϕ_{xxx} + O(k^2) + aO(h^3)
\end{align*}$$ So, this is consistent.
④ leap frog
$$\begin{align*} P_{h,k} ϕ &= \frac{ϕ^+ - ϕ^-}{2k} + a\frac{ϕ_+ -ϕ_-}{2h} \\ &= ϕ_t + \frac{k^2}6 ϕ_{ttt} + aϕ_x + a\frac{h^2}6 ϕ_{xxx} + O(k^3) + aO(h^3) \end{align*}$$ 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, $$ \frac1k \left\{ kϕ_t + \frac{k^2}2ϕ_{tt} + O(k^3) - \frac{h^2}2ϕ_{xx} + O(h^3)\right\} \\ =ϕ_t + \frac k2 ϕ_{tt} + O(k^2) - \frac{h^2}{2k}ϕ_{xx} + \frac{h^2}k O(h) $$ 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-Nicolson
$$ P_{k,h}ϕ = \frac{v^+ - v}k + a \frac{v^+_+ - v^+_- + v_+ - v_-}{4h} = 0 $$ Refer (4.1) $$ v_+^+ - v_-^+ = \ 2hϕ_x + 2khϕ_{xt} + k^2h ϕ_{ttx} + O(k^3h^3) \tag9$$ and $$ v_\pm = ϕ\pm hϕ_x + \frac{h^2}2ϕ_{xx} + O(h^3) $$ then, $$v_+ - v_- = 2hϕ_x + O(h^3)\tag{10}$$ 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, (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.