 Solution

Let $h=\Delta x$ and sample a sufficiently smooth periodic exact solution. The central finite differences have expansionsandThe 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 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.