= Solution
Let $h=\Delta x$ and sample a sufficiently smooth periodic exact solution. The <central finite differences> have expansions
$$
\frac{u(x-h)-2u(x)+u(x+h)}{h^2}
=u_{xx}+\frac{h^2}{12}u_{xxxx}+O(h^4),
$$
and
$$
\frac{u(x+h)-u(x-h)}{2h}
=u_x+\frac{h^2}{6}u_{xxx}+O(h^4).
$$
The semidiscrete right side therefore equals $u_{xx}+\alpha u_x+h^2(u_{xxxx}/12+\alpha u_{xxx}/6)+O(h^4)$. The spatial <consistency of a numerical method> is \b[second order], for each fixed $\alpha$. There is no time discretization in the stated ordinary differential system, so no separate temporal approximation order is being assigned. The Taylor statement concerns smooth solutions; <L2-compatible initialization of grid data> handles nonsmooth initial data without undefined point sampling.
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