= Solution
Put $h=\Delta x$. The spatial operator is the sum of a centred $x$ difference and a <dissipative second-order forward advection stencil> in $y$. For a smooth function, their <Taylor expansions> give
$$
\frac{u(x+h,y)-u(x-h,y)}{2h}
=u_x+\frac{h^2}{6}u_{xxx}+O(h^4),
$$
$$
\frac{-3u(x,y)+4u(x,y+h)-u(x,y+2h)}{2h}
=u_y-\frac{h^2}{3}u_{yyy}-\frac{h^3}{4}u_{yyyy}+O(h^4).
$$
Thus the leading spatial <local truncation error> is $h^2(u_{xxx}/6-u_{yyy}/3)$, generally nonzero. \b[The semidiscretization is second order in space.] Time has not yet been discretized, so there is no separate time order or time-step condition at this stage.
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