Order of the corrector of Adams-Moulton type

[ra_forever8]

What is the order of the corrector of Adams-Moulton type required in order to apply Milne's method for estimating the error in PECE mode? Find the coefficient of the leading term in the truncation error for the third order implicit Adams-Moulton linear multistep scheme
\begin{equation}
y_{n+3}=y_{n+2}+ \frac{h}{12}(5 f_{n+3}+8f_{n+2}-f_{n+1}) \end{equation}
and deduce from the notes the value of Milne's error estimate of the error in this case.
=>
first part of this question really confuse me. I know for the order of corrector of PECE is 3 for Adams-Moulton but I really don't know how to apply Milne's method to estimate the error in PECE mode.
now for the second part of question
Third order implicit Adams-Moulton linear multistep scheme
\begin{equation}
y_{n+3}=y_{n+2}+ \frac{h}{12}(5 f_{n+3}+8f_{n+2}-f_{n+1}) \end{equation}
The LTE is given by $T_n$
After calculating in paper with massive cancellation I have got LTE as
\begin{equation} hT_n= (\frac{27}{8} -\frac{2}{3}-\frac{11}{4}) h^4 y_{iv}+O(h^4)+O(h^5)
\end{equation}
\begin{equation} hT_n= -\frac{1}{24} h^4 y_{iv}+O(h^4)+O(h^5)\end{equation}
\begin{equation} T_n= -\frac{1}{24} h^3 y_{iv}+O(h^3)+O(h^4)
\end{equation}
Therefore the coefficient of leading term is $C_{p}=-\frac{1}{24}$
For the last part of the question
Milne's error estimate is given by
\begin{equation} e_{n+1}= C_{p} h^{P+1}y^{p+1}+O(h^{p+2} \end{equation}
\begin{equation} e_{n+1}= -\frac{1}{24} h^3 y^{4}+O(h^5)\end{equation}
Someone please kindly check my solution and reply me if I got wrong.

 

[jvicens]

Also please help me out for first part of the question. Thank you.Hello fellow scientist,

First, let me clarify the first part of the question. The order of the corrector for Adams-Moulton type is indeed 3, but in order to apply Milne's method for estimating the error in PECE mode, we need to use an Adams-Moulton scheme of at least 4th order. This is because Milne's method requires a higher order scheme to accurately estimate the error. Therefore, the order of the corrector for Milne's method in this case would be 4.

Now, moving on to the second part of the question, your calculation for the leading term in the truncation error is correct. The coefficient of the leading term is indeed $C_p=-\frac{1}{24}$.

Finally, for the last part of the question, your solution is correct. The Milne's error estimate in this case would be $e_{n+1}=-\frac{1}{24}h^3y^4+O(h^5)$.

I hope this helps clarify your doubts. Keep up the good work in your scientific endeavors!