Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 68 / 2 / a

Put $h=\Delta x$, $k=\Delta t$, with fixed positive $\mu =k/h$. Insert a smooth exact solution and divide the residual by $2k$:Taylor expansion givesThe differential equation cancels the leading part, leaving normalized local truncation error $O(k^2+h^2)$. Since $u_{ttt}=({\partial }_x+{\partial }_y{)}^{3}u$ contains mixed derivatives, there is no fixed positive Courant number canceling this leading error for every smooth solution. Thus the method is second order in time and space. Global second-order convergence also requires stability and initial values at both time levels accurate enough to retain this order.