 Fourth-order two-step advection stencil

For $u_t=u_x$, the recurrence $U_m^{n+1}-U_m^{n-1}={\sum }_{j=\pm 1,\pm 2}{a}_j{U}_{m+j}^n$ has fourth-order normalized consistency of a numerical method at fixed $\mu =k/h$ ifTo derive this, match moments $M_r={\sum }_j{a}_j{j}^r$ to the temporal Taylor series: $M_0=M_2=M_4=0$, $M_1=2\mu$, $M_3=2{\mu }^3$. The even conditions force antisymmetry; the two odd conditions then determine the displayed coefficients. The normalized residual begins with $h^4({\mu }^2-1)({\mu }^2-4)u_{xxxxx}/120$. Its amplification polynomial of a multilevel finite difference scheme is $G^2-2iA(\theta )G-1$, where $A=a_1\sin \theta +a_2\sin 2\theta$. At $\mu =1/2$, $|A|\le 3/4$, giving a uniform power bound. At $\mu =3/2$, $A(\pi /3)=19\sqrt{3}/32>1$, and wave packets on the growing polynomial root prove Cauchy linear instability.