  Before proving (unconditional) energy stability for gradient flows, most
existing studies either require a strong Lipschitz condition regarding the
non-linearity or certain $L^{\infty}$ bounds on the numerical solutions (the
maximum principle). However, proving energy stability without such premises is
a very challenging task. In this paper, we aim to develop a novel analytical
tool, namely global-in-time energy stability, to demonstrate energy dissipation
without assuming any strong Lipschitz condition or $L^{\infty}$ boundedness.
The fourth-order-in-space Swift-Hohenberg equation is used to elucidate the
theoretical results in detail. We also propose a temporal second-order accurate
scheme for efficiently solving such a strongly stiff equation. Furthermore, we
present the corresponding optimal $L^2$ error estimate and provide several
numerical simulations to demonstrate the dynamics.
