Verlet method global error

Question

I was trying to understand the global error calculation for the verlet method on Wikipedia but it's not so clear to me when it goes from:

to

Shouldn't the error relative to x'' be considered as well?

Answer 1

Yes, it should be. I think that "proof" on Wikipedia is bad and isn't sufficient. It is right that the global error of Verlet is $\mathcal{O}(\Delta t^2)$ but it is no the way to show this. You can find some information in
Vesely F. J., Computational Physics: An Introduction (2nd edition ed.). 2001, page 105.

Or you can take another approach. By using some transformation you will see something interesting. Look, I will start on Verlet, let $h = \Delta t$ and $x_i = x(t_i) = x(t_0 + ih)$. We get \begin{align} x_{i+1} & = 2x_i - x_{i-1} + h^2 f(x_i) \\[10pt] & = x_i + x_i - x_{i-1} + h^2 f(x_i) \\[10pt] & = x_i + \frac{x_i - x_{i-1}}{h}h + h^2f(x_i) \\[10pt] & = x_i + hx'_i + h^2f(x_i) \\[10pt] & = x_i + h(x'_i +hf(x_i)) \end{align} and this leads to $$ \left\lbrace \begin{array}{ll} x'_{i+1} = x'_i + hf(x_i) \\[6pt] x_{i+1} = x_i + hx'_{i+1} \end{array} \right. $$ which is the Euler-Cromer method. Now you can investigate what happened with global error in this scheme. Treat this like a clue. I am not 100% sure that these two methods are close connected. But I made some calculations and I got the same values on $x$ in both.