= Compact third-order advection stencil
{title2=$K(\mu)=\mu(\mu+1)(\mu-1)(\mu-2)$}
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 $aU_{j-1}^{n+1}+bU_j^{n+1}+cU_{j+1}^{n+1}=dU_j^n+eU_{j+1}^n$ 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.
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