= Solution
\b[The original PDF has a spatial shift $u_{m+2}^{n+2}$, so the printed scheme is not the usual second-order BDF discretization.] The PDF does have $-2u_m^{n+2}$ on the right; the TeX aid loses that factor two, and that transcription must not be used.
Let $h=\Delta x$, $k=\Delta t$, and expand an exact smooth <heat equation> solution about $(x_m,t_{n+2})$. Relative to the standard unshifted <backward differentiation formula>, the printed newest value adds
$$
u(x_m+2h,t)-u(x_m,t)=2hu_x+2h^2u_{xx}+O(h^3).
$$
Dividing the raw residual by $2k/3$, its expansion is
$$
u_t-u_{xx}+\frac{3h}{k}u_x+\frac{3h^2}{k}u_{xx}
-\frac{k^2}{3}u_{ttt}-\frac{h^2}{12}u_{xxxx}
+O(k^3+h^4+h^3/k).
$$
In particular, for the usual refinement $k=\mu h^2$ with fixed $\mu>0$, the term $3u_x/(\mu h)$ generally diverges. A concrete smooth counterexample is $u=e^{-\pi^2t/4}\cos(\pi x/2)$, which satisfies the <Dirichlet boundary conditions> and has $u_x\ne0$ at $x=1/2$. Thus \b[the printed scheme has no consistent order under fixed-$\mu$ refinement]. Its normalized shift error is $O(h/k)$, so one could only investigate a special coupled limit such as $h=o(k)$ after specifying a valid boundary closure; it has no standard joint second-order accuracy.
For the likely intended correction $u_m^{n+2}$, the extra shift terms disappear. The normalized <local truncation error> is
$$
-\frac{k^2}{3}u_{ttt}-\frac{h^2}{12}u_{xxxx}+O(k^3+h^4).
$$
With a stable, suitably accurate starter, the corrected <backward differentiation formula> combined with the <second-order central difference> therefore has
$$
\boxed{\text{order two in time and order two in space}}.
$$
This is a correction to the printed method, not a claim about it.
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