= Solution
Divide the recurrence by $2k$ and substitute a smooth exact solution. Centered <Taylor expansions> in time and space give
$$
\frac{u(t+k)-u(t-k)}{2k}-\frac{u(x+\delta,y)-u(x-\delta,y)+u(x,y+\delta)-u(x,y-\delta)}{2\delta}
=\frac{k^2}{6}u_{ttt}-\frac{\delta^2}{6}(u_{xxx}+u_{yyy})+O(k^4+\delta^4).
$$
Thus the <leapfrog advection scheme> is consistent with normalized residual $O(k^2+\delta^2)$. Its unnormalized defect is $O(k^3+k\delta^2)$. For a fixed $0<\nu<1/2$, the uniform two-level bound proved above and accumulation over $O(1/k)$ steps yield, for a sufficiently smooth solution,
$$
\max_{nk\leq T}\|e^n\|_{2,\delta}\leq C_{T,\nu}\bigl(\|e^0\|_{2,\delta}+\|e^1\|_{2,\delta}+k^2+\delta^2\bigr).
$$
Both starting levels must be supplied consistently; errors of order $k^2+\delta^2$ give second-order convergence. For general <square-integrable functions> as initial data, point sampling is not defined on equivalence classes. Use cell averages or another bounded grid projection, together with a bounded consistent startup. Approximate the initial data by smooth functions, apply the smooth-data convergence estimate, and use the uniform numerical and exact translation bounds to pass to the <L2 space> limit. This is the content of <Lax equivalence theorem> in the present setting.
At $\nu>1/2$ the growing <Fourier mode> prevents general convergence. At the endpoint, initial data supported in increasingly narrow neighborhoods of the double root can have a second-level error of norm $O(k)$ but a final error of order one after $O(1/k)$ steps. These are vanishing starting errors, so the endpoint fails general two-level convergence as well. Consequently
$$
\boxed{0<\Delta t/\Delta x<1/2}
$$
is also the general convergence range, with a consistent stable initialization. Claims for a particular specially chosen endpoint startup would be a different, restricted assertion.
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