Compact third-order advection stencil ($K(\mu )=\mu (\mu +1)(\mu -1)(\mu -2)$)

Let $a=\mu (1+\mu )/2$, $b=(1+\mu )(2-\mu )$, $c=(1-\mu )(2-\mu )/2$, $d=2-\mu$, and $e=1+\mu$. The implicit stencil $a{U}_{j-1}^{n+1}+b{U}_j^{n+1}+c{U}_{j+1}^{n+1}=d{U}_j^n+e{U}_{j+1}^n$ for $u_t=u_x$ has centered characteristic moments zero through degree three and fourth moment $K(\mu )$. Its Fourier numerator and denominator satisfy $|L(\theta ){|}^2-|N(\theta ){|}^2=K(\mu )(1-\cos \theta {)}^2$. The marching operator is a contraction exactly for nonnegative Courant numbers in $[0,1]\cup [2,\infty )$; with signed Courant numbers, also include $(-\infty ,-1]$. At the four roots of $K$ the method becomes an exact integer grid translation rather than merely third order.