Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 66 / 2 / a / Solution

The characteristic polynomials of a linear multistep method areTo determine formal order, substitute a smooth exact solution and expand about the first time level. The exponential-symbol order criterion for a multistep method collects precisely the same coefficients:The constant, linear and quadratic coefficients vanish for every $a$. The cubic coefficient vanishes only at $a=-1/5$, where the quartic coefficient is $-1/30\ne 0$. HenceHere order means the exact-solution step residual is $O(h^{p+1})$. It is a formal consistency result, not a convergence assertion. In particular at $a=1$ both ${\rho }^{\prime }(1)$ and $\sigma (1)$ vanish, and the double root at one destroys zero-stability; cancelling its common factor gives a different, first-order recurrence with an additional integration constant left unspecified by the original formula.