The global in time energy estimate is derived for the  ETDRK2 numerical scheme for the phase field crystal (PFC) equation,  a sixth order parabolic equation to model crystal evolution.  The energy stability is available for the exponential time  differencing Runge-Kutta (ETDRK) numerical scheme to the  gradient flow equation, under an assumption of global Lipschitz  constant. To recover the stabilization constant value, some  local-in-time convergence analysis has been reported, so that  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, so that the energy dissipation property is valid 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, turns out to be 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, so that the stabilization parameter  value could be theoretically justified. This justification in turn  ensures the energy dissipation at the next time step, so that the  mathematical induction could be effectively applied, and the  global-in-time energy estimate is accomplished. This technique  is expected to be available for many other Runge-Kutta numerical schemes.

12.25.pdf