 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 residualThe constant, linear and quadratic Taylor coefficients vanish. Its cubic coefficient isFor 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 isHence 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.