Past exam of the numerical linear algebra and numerical solution of differential equations course of the University of Oxford / 2020 / 2 / b

Fix the following normalization of the local truncation error, also called the consistency error: insert a smooth exact solution into the linear multistep method and divide the residual by $h$,Thus the unscaled defect is $h{T}_n(h)$. Consistency of a numerical method means that $T_n(h)\to 0$ uniformly over mesh points in each fixed interval for every sufficiently smooth exact solution. Write the characteristic polynomials of a linear multistep method asBy Taylor's theorem, consistency of a numerical method is equivalent toor $\rho (1)=0$ and ${\rho }^{\prime }(1)=\sigma (1)$.

Second-order accuracy means $T_n(h)=O(h^2)$ for smooth solutions, equivalently an unscaled local truncation error $O(h^3)$. In addition to the two consistency conditions, its order conditions requireAn order of a numerical method of exactly two additionally requires that the next order condition fails. These statements concern the local defect; a second-order global error requires stability and suitably accurate starting values as well.