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I am in numerical methods and my professor introduced how one may construct a higher order method for multi-step methods for $y'(x) = f(x,y(x))$ in the form:

$\sum_{s=0}^{k}\alpha_s y_{n+s} = h\sum_{s=0}^{k}\beta_s f(x_{n+s}, y_{n+s})$

As:

  1. We look at the local truncation error $(T_h)_n = \frac{1}{h}\sum\alpha_s y_{n+s}-\sum_{s=0}^{k}\beta_s f(x_{n+s}, y_{n+s})$
  2. Define the linear map $(L_h y)_n = \frac{1}{h}\sum\alpha_s y(x+sh)-\sum_{s=0}^{k}\beta_s y'(x+sh)$
  3. Find $\alpha$'s and $\beta$'s such that $(L_h y)_n = 0$ for $y$ a polynomial.

Can someone give me a specific example (or reference to a written out example) of how you would do this? Thank you, in advance, for your time and help.

Bernard
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Reid
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    In https://math.stackexchange.com/questions/3473300/ I gave an algorithm to compute coefficients for Adams-Bashford-Moulton of any order. For BDF a similar mechanism exists. – Lutz Lehmann Mar 12 '21 at 19:48
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    https://math.stackexchange.com/questions/3358496/numerical-method-forward-finite-difference-coefficient is for forward difference quotients, BDF has reversed coefficient sequences with alternating signs. The general approach with the Taylor shift operator generalizes the usual manipulations of order conditions. – Lutz Lehmann Mar 13 '21 at 11:19
  • Thank you, Lutz! – Reid Mar 13 '21 at 13:23

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