Leapfrog advection scheme
= Leapfrog advection scheme
Centered space and leapfrog time discretization of $u_t+cu_x=0$ gives
$$
u_j^{n+1}=u_j^{n-1}-\nu(u_{j+1}^n-u_{j-1}^n),
\qquad \nu=c\Delta t/\Delta x.
$$
Its amplification polynomial is $G^2+2i\nu\sin\theta\,G-1$. The two roots have unit modulus for $|\nu\sin\theta|\leq1$, but a repeated unit root at equality violates the uniform root condition.