 Two-dimensional leapfrog stability threshold

For $u_t=u_x+u_y$ with equal spatial mesh sizes, the centered leapfrog recurrence has amplification polynomial of a multilevel finite difference scheme $G^2-2i\mu (\sin \xi +\sin \eta )G-1$. Uniform stability for arbitrary two-level starting perturbations holds exactly for $0<\mu <1/2$. At $\mu =1/2$ the phase pair $(\pi /2,\pi /2)$ gives $(G-i{)}^2$, and its companion matrix has a nontrivial Jordan block, producing growth proportional to the number of steps. For larger $\mu$ one polynomial root has modulus greater than one. Bounds on eigenvalue moduli alone therefore give a misleading non-strict endpoint.