= Fourth-order two-step advection stencil
{title2=$a_1=\mu(4-\mu^2)/3,\quad a_2=\mu(\mu^2-1)/6$}
For $u_t=u_x$, the recurrence $U_m^{n+1}-U_m^{n-1}=\sum_{j=\pm1,\pm2}a_jU_{m+j}^n$ has fourth-order normalized <consistency of a numerical method> at fixed $\mu=k/h$ if
$$
a_1=\mu(4-\mu^2)/3,\quad a_2=\mu(\mu^2-1)/6,\quad a_{-j}=-a_j.
$$
To derive this, match moments $M_r=\sum_ja_jj^r$ 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 $A=a_1\sin\theta+a_2\sin2\theta$. At $\mu=1/2$, $|A|\le3/4$, giving a uniform power bound. At $\mu=3/2$, $A(\pi/3)=19\sqrt3/32>1$, and <wave packets> on the growing <polynomial root> prove Cauchy <linear instability>.
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