Derivation of adams-moulton determine the coefficients

Question

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

τ^n+r = (1/Δt) (_j=0∑^r α_ju(t_n+j)- Δt _j=0∑^r β_j f(u(t_n+j)).|

Note that α₀ = 0| α₁ = -1| and α₂ = 1|

(b) Using the relation

u(t_n+2) = u(t_n+1) + _{t_n+1}∫^t_n+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(U^n+j)|. It's easiest to use the "Newton form" of the interpolating polynomial and consider the three times t_n = -Δt|t_n+2 = 0|, and t_n+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.)