= Implicit advection scheme with exact integer shifts
{title2=$\mu\in(-\infty,-1]\cup[0,1]\cup[2,\infty)$}
A three-point implicit advection stencil can match translated data through cubic <Taylor expansion> while retaining a rational <Fourier amplification symbol>. For the coefficient family $a=\mu(1+\mu)/6$, $b=(2-\mu)(1+\mu)/3$, $c=(2-\mu)(1-\mu)/6$, $d=(2-\mu)/3$, $e=(1+\mu)/3$, its factor is $G=(d+e e^{i\theta})/(a e^{-i\theta}+b+c e^{i\theta})$. Direct subtraction gives $|D|^2-|N|^2=\mu(\mu-2)(\mu-1)(\mu+1)(1-\cos\theta)^2/9$. The displayed ranges are the contraction ranges in the <discrete L2 norm>, with a nonvanishing denominator there. For forward time and positive speed parameter, retain the nonnegative ranges. For fixed nonzero nonexceptional <Courant number>, the normalized truncation error is third order; at $\mu=-1,0,1,2$ the update is the corresponding exact grid translation.
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