Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 39A / i / Solution

Put $\tau =\Delta t$ and $h=\Delta x$, with $h^2=\tau /\mu$ for fixed $\mu >0$. Substitute a smooth exact solution of the heat equation into the Adams-Bashforth method. Define its one-step defect $R$ as the left side minus the right side of the unscaled update. The centered finite difference satisfiesThe Taylor expansion in time givesSince $u_t=u_{xx}$ and $u_{tt}=u_{xxt}$, the first two terms cancel, leavingThis is the requested local error of the unscaled update. If “local truncation error” denotes the defect divided by $\Delta t$, it is instead $O(\Delta t)$ under fixed $\mu$: the centered spatial error is then $O(h^2)=O(\Delta t)$. The scheme is second order in time at fixed spatial discretization, which is a separate order statement.