Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 18D / b

Insert an exact solution into the linear multistep method and use $f(t,y(t))=y^{\prime }(t)$. Its local residual isExpanding both terms by Taylor theorem givesConsequently $C_0=\cdots =C_p=0$ is sufficient for order at least $p$. It is also necessary: test the residual on the polynomials $y(t)=1,t,\dots ,t^p$, evaluating at $t=0$. For $y(t)=t^k$, it equals $h^k{C}_k$ (and equals $C_0$ for the constant), which can be $O(h^{p+1})$ only if $C_k=0$. These functions can be realized as exact solutions of suitable smooth ordinary differential equations, so this is a genuine necessity test. Thus the order conditions areThis is a statement about local order; zero-stability is the separate condition that turns it into convergent approximation.