= Solution
The original PDF puts both neighbouring values at time $n+1$. Thus the actual scheme is <Backward Euler method> in time with a centered three-point approximation to the second spatial derivative. The old-time neighbours in the TeX transcription describe a different method.
Let $h=\Delta x$ and $k=\Delta t$. Insert a sufficiently smooth exact solution and divide by $k$. Expanding at $(x_m,t_{n+1})$ gives
$$
\frac{u(x_m,t_{n+1})-u(x_m,t_n)}k
-\frac{u(x_m-h,t_{n+1})-2u(x_m,t_{n+1})+u(x_m+h,t_{n+1})}{h^2}
=-\frac{k}{2}u_{tt}-\frac{h^2}{12}u_{xxxx}+O(k^2+h^4).
$$
The PDE cancels the $u_t-u_{xx}$ term. Therefore the normalized <local truncation error> is $O(k+h^2)$ and
$$
\boxed{\text{first order in time and second order in space}.}
$$
For compatible smooth data with exact nodal initialization, the <stability of a numerical method> established below gives <global error> $O(k+h^2)$ on fixed time intervals, by summing the one-step defects with the contraction bound. Thus no relation between $k$ and $h$ is needed for stability; choosing $k=O(h^2)$ makes both consistency contributions $O(h^2)$.
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