0914 schemes’ consistency and stability

Let, \( v^+ = v_m^{n+1}, v_- = v_{m-1}^n \) and so on.

And usually, \(v = \phi\)

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} \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*}

$$

If the PDE is \( u_t + au_x = 0 \)

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, $$ P_{h,k} - Pϕ = \frac k2ϕ_{tt} + a\frac h2ϕ_{xx} + O(k^2) + aO(h^2) \tag6$$ 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 \\

 &= ϕ_t + \frac k2ϕ_{tt}  + aϕ_x + a\frac h2ϕ_{xx} + O(k^2) + aO(h^2) \tag{5.1}

\end{align*}$$