Amplification polynomial of a multilevel finite difference scheme

For a schemethe Fourier ansatz gives the amplification polynomialIts roots are the amplification factors of that Fourier mode.

Centered space and leapfrog time discretization of $u_t+c{u}_x=0$ givesIts amplification polynomial is $G^2+2i\nu \sin \theta G-1$. The two roots have unit modulus for $|\nu \sin \theta |\le 1$, but a repeated unit root at equality violates the uniform root condition.

Applying a centered leapfrog step in time and the centered second difference in space to $u_t=u_{xx}$ givesFor every $\mu >0$, each nonconstant Fourier mode has an amplification root of modulus greater than one, so the scheme is unconditionally unstable.