Predictor-corrector scheme $S^{\alpha}_{\beta}$ for a hyperbolic equation of conservation For the following hyperbolic equation of conservation:
$$\frac{\partial f}{\partial t}+\frac{\partial G(f)}{\partial x}=0$$
I need your help to understand how we derive the following scheme so-called $S^{\alpha}_{\beta}$
\begin{aligned}
& \tilde{f}_{i}=(1-\beta) f_{i}^{n}+\beta f_{i+1}^{n}-\alpha \frac{\Delta t}{\Delta x}\left(G_{i+1}^{n}-G_{i}^{n}\right) \\
& f_{i}^{n+1}=f_{i}^{n}-\frac{\Delta t}{2 \alpha \Delta x}\left[(\alpha-\beta) G_{i+1}^{n}+(2 \beta-1) G_{i}^{n} + (1-\alpha-\beta) G_{i-1}^{n}+\tilde{G}_{i}-\tilde{G}_{i-1}\right]
\end{aligned}
where $\tilde{G}_{i} = G( \tilde{f}_{i})$.
Also, I need to know how we can calculate the amplification matrix, the modified equation, and how we could find optimal values of $\alpha$ and $\beta$ to get a better approximation.
Any readings, insights, examples, are most welcome. Thanks for your help.
 A: This is a predictor-corrector scheme of a similar kind than the Richtmyer and MacCormack methods (see this article and references therein). Note that if $\alpha = 1$ and $\beta = 0$, then
\begin{aligned}
\tilde{f}_{i} &= f_{i}^{n}-\tfrac{\Delta t}{\Delta x}\left(G_{i+1}^{n}-G_{i}^{n}\right) \\
f_{i}^{n+1} &=f_{i}^{n}-\tfrac{\Delta t}{2 \Delta x}\big[ G_{i+1}^{n}- G_{i}^{n} +\tilde{G}_{i}-\tilde{G}_{i-1}\big] \\
&= \tfrac12( f_{i}^{n} + \tilde f_i ) -\tfrac{\Delta t}{2 \Delta x}\big[ \tilde{G}_{i}-\tilde{G}_{i-1}\big]
\end{aligned}
gives exactly the MacCormack method. If $\alpha = 1$ and $\beta= \frac12$, then
\begin{aligned}
\tilde{f}_{i} &= \tfrac12(f_{i}^{n} + f_{i+1}^{n})-\tfrac{\Delta t}{\Delta x}\left(G_{i+1}^{n}-G_{i}^{n}\right)\\
f_{i}^{n+1} &=f_{i}^{n}-\tfrac{\Delta t}{2 \Delta x}\big[\tfrac12 G_{i+1}^{n}- \tfrac12 G_{i-1}^{n} +\tilde{G}_{i}-\tilde{G}_{i-1}\big] \\
&= \tfrac12\left( f_{i}^{n} -\tfrac{\Delta t}{2 \Delta x}\big[ G_{i+1}^{n}- G_{i-1}^{n}\big] + f_{i}^{n} -\tfrac{\Delta t}{\Delta x}\big[\tilde G_{i+1}- \tilde G_{i}\big] \right)
\end{aligned}
gives an average of the Richtmyer and explicit centered methods. Therefore, the present method is a variation of these methods, with linear averages and weighted extrapolations.
Now, consider the case where $G(f) = af$ is linear with slope $a>0$. We find by substitution that
$$
f_i^{n+1} = f_i^n - \frac{a \Delta t}{2 \Delta x} \left( f_{i+1}^n - f_{i-1}^n\right) + \frac{a^2 \Delta t^2}{2\alpha \Delta x^2}(f_{i+1}^n - 2f_{i}^n + f_{i-1}^n) \, .
$$
A general centered time-stepping formula is recovered, where the numerical viscosity term (second-order centered spatial difference) is divided by $\alpha$. If $\alpha = 1$, $a \, {\Delta t}/{\Delta x}$ or $a^2$, then the Lax-Wendroff, upwind scheme and Lax-Friedrichs methods are obtained, respectivelly.
