Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 18C (Numerical Analysis)

Insert the exact solution into the linear multistep method and expand about $t_n$. Its defect isThe coefficient of $h^k{y}^{(k)}(t_n)$ isfor $k\ge 1$, while the constant coefficient is ${\sum }_l{\rho }_l$. The method has order at least $p$ exactly when these coefficients vanish for $0\le k\le p$.

On the other hand,The coefficient of $z^k$ in this expression is precisely the preceding order-condition coefficient. Therefore all coefficients through degree $p$ vanish exactly whenwhich proves the equivalence.

For this Adams–Moulton method, the characteristic polynomials of a linear multistep method areExpansion at zero givesPart (a) therefore shows that the method has order exactly three.

It is consistent because its order is at least one. The roots of $\rho (w)=w(w-1)$ are $0$ and $1$; both lie in the closed unit disk, and the only unit-modulus root is simple. Thus the method is zero-stable. By the Dahlquist equivalence theorem, consistency and zero stability imply convergence. Hence the method is