= Two-dimensional leapfrog stability threshold
{title2=$0<\mu<\tfrac12$}
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.
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