= Rational implicit advection stencil with exact shift exceptions
{title2=$\mu=\Delta t/\Delta x$}
Consider a one-step stencil with new-level coefficients $a=\mu(1+\mu)/2$, $b=(1+\mu)(2-\mu)$ and $c=(1-\mu)(2-\mu)/2$ at offsets $-1,0,1$, and old-level coefficients $d=2-\mu$, $e=1+\mu$ at offsets $0,1$. For $u_t=u_x$, its exact-solution residual starts at fourth degree with coefficient $\mu(\mu-2)(\mu-1)(\mu+1)/24$. Thus its normalized <local truncation error> is third order at fixed nonzero <Courant number>, except at the exact-shift values $-1,1,2$; zero gives the zero-step identity. For its <Fourier symbol>, $|D|^2-|N|^2=\mu(\mu-2)(\mu-1)(\mu+1)(1-\cos\theta)^2$. Uniformly invertible stable steps occur for $\mu\leq-1$, $0\leq\mu\leq1$ or $\mu\geq2$.
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