Implicit advection scheme with exact integer shifts ($\mu \in (-\infty ,-1]\cup [0,1]\cup [2,\infty )$)

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.