Local Truncation Error of Adam Bashforth 3-step I want to get the local truncation error of this Adam Bashforth 3-step method
$$y_{n+1} = y_n + \frac{h}{12} (23 f_{n+2} − 16 f_{n+1} + 5 f_n)$$
I started by remembering the following Taylor development
$$ y(x_{n+1}) = y(x_n) +hy'(x_n) +\frac{h^2}{2}y''(x_n)+\frac{h^3}{6}y'''(x_n)+\frac{h^4}{24 }y^{(4)}(x_n)$$
After, I do the subtraction of these two expressions and I get the following
$$y(x_{n+1})- y_{n+1} = y(x_n)-y_n + hy'(x_n) - \frac{h}{12} (23 f_{n+2}) +\frac{h^2}{2}y''(x_n)+ 16\frac{h}{12} (f_{n+1}) +\frac{h^3}{6}y'''(x_n)- 5\frac{h}{12} (f_n) +\frac{h^4}{24 }y^{(4)}(x_n)$$
But from here I don't know how to continue since I try to simplify some terms to finally arrive at $$LTE=\frac{h^4}
{8} y^{(4)}(\eta).$$
I'm struggling to catch the error in the calculation but I'm finally stuck
 A: The method with the definition you gave takes into account future points and does not have an $\mathcal{O(h^4)}$ local error. If you modify the definition of the method to take into account only past points
$$y_{n+1}-y_n=\frac{h}{12}(23f_n-16f_{n-1}+5f_{n-2})$$
where $f_k:=f(x_k,y_k)$ with $x_k=x_0+kh, y_k=y(x_k), k\leq n$. To compute the local error we just need to expand around the point $x_n$. To this end we first compute the Taylor estimate as you did
$$y(x_{n}+h)-y(x_n)=hy'_n+\frac{h^2}{2}y''_n+\frac{h^3}{6}y'''_n+\frac{h^4}{24}y''''_n +\mathcal{O}(h^5)$$
and then we can also compute an estimate $y_{n+1}$ of $y(x_n+h)$ by the method. For this calculation we need to Taylor expand $f_{n-1}$ and $f_{n-2}$ as follows:
$$f_{n-1}:=f(x_n-h, y(x_n-h))=y'(x_n-h)=y'(x_n)-hy''(x_n)+\frac{h^2}{2}y'''(x_n)-\frac{h^3}{6}y''''(x_n)+\mathcal{O}(h^4)$$
$$f_{n-2}:=f(x_n-2h, y(x_n-2h))=y'(x_n-2h)=y'(x_n)-2hy''(x_n)+{2h^2}y'''(x_n)-\frac{8h^3}{6}y''''(x_n)+\mathcal{O}(h^4)$$
We see that the Taylor estimate for $y_{n+1}$ gives
$$y_{n+1}-y_n=hy'_n+\frac{h^2}{2}y''_n+\frac{h^3}{6}y'''_n-\frac{h^4}{3}y''''_n +\mathcal{O}(h^5)$$
Subtracting then two equations and taking an absolute value gives that the local error can be estimated as
$$|y(x_n+h)-y_{n+1}|\leq\frac{3M}{8}h^4$$
where $M=\sup_{\xi\in (x_n-2h,x_n+h)}|y^{(4)}(\xi)|$. This constant $M$ certainly exists if all partial derivatives of $f(x,y)$ of the 4-th order exist and are continuous.
