Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 69 / 2 / b

Factor the first characteristic polynomial:The unit root is simple and the other root is strictly inside the unit disk, so the method is zero-stable. Also $\rho (1)=0$ and ${\rho }^{\prime }(1)=\sigma (1)=2/7$, giving numerical consistency. The Dahlquist equivalence theorem therefore makes it convergent.

The mechanism can also be seen directly. The zero-stable homogeneous recurrence has bounded propagators because its two error modes are $1$ and $(5/7{)}^n$. Variation of constants for the perturbed recurrence bounds the error by a fixed multiple of its two starting errors and the sum of the step defects, plus $O(h)$ times a sum of errors from the Lipschitz vector field. Absorb the current implicit error term for sufficiently small $h$, then apply the discrete Gronwall inequality. On a fixed time interval the sum of $O(h^3)$ exact-start defects is $O(h^2)$, giving second-order numerical convergence with second-order starting values. The implicit step is locally well defined, for example by contraction when $4hL/7<1$ for a Lipschitz constant $L$.