= Solution
Write $h=\Delta x$, $k=\Delta t=\mu h$. Every smooth exact solution of the <advection equation> here is $u(x,t)=F(x+t)$. At $s=x+t$, substitution into the scheme gives the unscaled residual
$$
\mathcal R=F(s+\mu h)-(1-2\mu)[F(s)-F(s+h)]-F(s+(1-\mu)h).
$$
The constant, linear and quadratic Taylor coefficients vanish. Its cubic coefficient is
$$
\frac{h^3}{6}\{\mu^3+1-2\mu-(1-\mu)^3\}F'''(s)
=\frac{\mu(1-\mu)(1-2\mu)}6h^3u_{xxx}.
$$
For a general smooth function the first-order residual is $2k(u_t-u_x)$, so the proper normalized <local truncation error> on an exact solution is
$$
\boxed{\frac{\mathcal R}{2k}=\frac{(1-\mu)(1-2\mu)}{12}h^2u_{xxx}+O(h^3).}
$$
Hence \b[the scheme has formal order two for fixed general positive $\mu$]. At $\mu=1/2$ and $\mu=1$, the displayed residual vanishes identically for every $F$, not just to one higher Taylor order: the recurrence transports exactly sampled characteristics when both starting levels are compatible with the exact solution. This formal exactness does not imply <stability>. At $\mu=0$ there is no positive time step. The two-level recurrence also requires a starting procedure for $U^1$; one initial solution profile alone is not two independent starting levels.
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