Reference no: EM131016829
Question 1 - Derivation of Adams-Moulton
Determine the coefficients β0, β1, β2| for the third order, 2-step Adams-Moulton method. We will do this two different ways:
(a) Using the general expression for the local truncation error for multi-step methods
τn+r = (1/Δt) (j=0∑r αju(tn+j)- Δt j=0∑r βj f(u(tn+j)).|
Note that α0 = 0| α1 = -1| and α2 = 1|
(b) Using the relation
u(tn+2) = u(tn+1) + t_n+1∫tn+2 f(u(s)) ds.|
Interpolate a quadratic polynomial p(t)| through the three values f(Un), f(Un+1) and f(Un+2)| and then integrate this polynomial exactly to obtain the formula. The coefficients of the polynomial will depend on the three valuesf(Un+j)|. It's easiest to use the "Newton form" of the interpolating polynomial and consider the three times tn = -Δt|tn+2 = 0|, and tn+2 = Δt| so that p(t)| has the form
p(t) = A + B(t + Δt) + C(t + Δt) t|
where A. B| and C| are the appropriate divided differences based on the data. Then integrate from 0| to Δt|. (The method has the same coefficients at any time. so this is valid.)
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