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Math Help - Numerical solution of differential equations

  1. #1
    Junior Member
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    Numerical solution of differential equations

    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.
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  2. #2
    Super Member
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    Re: Numerical solution of differential equations

    You might get a better response in the differential equations forum - your question is pretty advanced for the calculus forum. That or my training in differential equations is lacking.

    - Hollywood
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