= Lax-Wendroff advection scheme
{c}
{title2=$|\nu|\leq1$}
For the <advection equation> $u_t+cu_x=0$ on a uniform grid, the Lax-Wendroff update is
$$
U_j^{n+1}=U_j^n-\frac\nu2(U_{j+1}^n-U_{j-1}^n)+\frac{\nu^2}2(U_{j+1}^n-2U_j^n+U_{j-1}^n),\qquad \nu=c\Delta t/\Delta x.
$$
Its <amplification factor> is $G=1-i\nu\sin\theta+\nu^2(\cos\theta-1)$, with $|G|^2=1-4\nu^2(1-\nu^2)\sin^4(\theta/2)$. Thus <Von Neumann stability analysis> on an infinite or periodic grid gives stability for $|\nu|\leq1$. The scheme has second <order of a numerical method> for sufficiently smooth solutions; finite-interval <boundary conditions> still need separate treatment.
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