Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 38E / a / Solution

Let all spatial derivatives be evaluated at $x=mh$, $t=nk$. Expand the two adjacent flux contributions symmetrically. The bracket in the scheme isFor clarity, the $h^2$ terms are $a{u}_{xx}+a^{\prime }{u}_x$; odd orders cancel under $h\mapsto -h$. Under four spatial derivatives the next coefficient is $a{u}_{xxxx}/12+a^{\prime }{u}_{xxx}/6+a^{\prime \prime }{u}_{xx}/8+a^{(3)}{u}_x/24$. Time expansion gives $u(x,t+k)=u+k{u}_t+O(k^2)$. Since $u_t=(a{u}_x{)}_x$ and $k=\mu h^2$,This stronger smooth-data residual implies the requested $O(h^3)$ consistency bound. Even a less detailed Taylor expansion with an $O(h^3)$ spatial remainder would suffice for the printed order. Uniform bounds on the needed derivatives over the region under consideration make the estimate uniform in the interior mesh index; the zero boundary values enter the scheme exactly.