Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 39C / a / Solution

Write $h=\Delta x$, $k=\Delta t$ and apply the recurrence to an exact smooth solution. The centred second difference is $h^2{u}_{xx}+h^4{u}_{xxxx}/12+O(h^6)$. Expanding the previous-time second difference and using $u_t=u_{xx}$ gives the one-step residualwhere $h^2=k/\mu$ with fixed positive $\mu$. Thus the local one-step error is $O((\Delta t{)}^2)$ as stated. If local truncation error is defined after division by the step, it is instead $O(\Delta t)$ in this coupled refinement; the spatial error is only second order in $h$. This distinguishes the printed local-error convention from the second-order time accuracy of the underlying Adams-Bashforth method at fixed spatial discretization.