  The global-in-time energy estimate is derived for the second-order accurate
exponential time differencing Runge-Kutta (ETDRK2) numerical scheme to the
phase field crystal (PFC) equation, a sixth-order parabolic equation modeling
crystal evolution. To recover the value of stabilization constant, some
local-in-time convergence analysis has been reported, and the energy stability
becomes available over a fixed final time. In this work, we develop a
global-in-time energy estimate for the ETDRK2 numerical scheme to the PFC
equation by showing the energy dissipation property for any final time. An a
priori assumption at the previous time step, combined with a single-step $H^2$
estimate of the numerical solution, is the key point in the analysis. Such an
$H^2$ estimate recovers the maximum norm bound of the numerical solution at the
next time step, and then the value of the stabilization parameter can be
theoretically justified. This justification ensures the energy dissipation at
the next time step, so that the mathematical induction can be effectively
applied, by then the global-in-time energy estimate is accomplished. This paper
represents the first effort to theoretically establish a global-in-time energy
stability analysis for a second-order stabilized numerical scheme in terms of
the original free energy functional. The presented methodology is expected to
be available for many other Runge-Kutta numerical schemes to the gradient flow
equations.
