A unified global-in-time energy stability analysis of ETDRK schemes for the Cahn–Hilliard–Ohta–Kawasaki equation

Abstract We conduct a comprehensive global-in-time energy stability of temporally first- to third-order accurate exponential-time-differencing Runge–Kutta schemes for the Cahn–Hilliard–Ohta–Kawasaki equation modeling the microphase separation of diblock copolymer melts. The theoretical challenge lies in the energy stability analysis without assuming global Lipschitz nonlinearity or an\ L^ bound on solutions. To address this, we propose a general framework for estimating a uniform bound on all stage solutions, which in turn determines the stabilization parameter required to ensure energy stability. This approach eliminates the need for traditional assumptions, as well as any restriction on the time step related to the spatial discretization or the convergence constant. Because of the absence of a high-order Sobolev norm in the energy functional, preliminary H^1 and H^2 estimates are established for each stage solution, with rough L^ bounds obtained via the log-interpolation-embedding inequality in two dimensions. Consequently, by constructing an inverse operator, we establish a uniform-in-time H^2 estimate for the final stage solution, subject to an O (^6 | |^-4) time-step constraint. This leads to a uniform-in-time L^ bound for all stage solutions through the log-interpolation-embedding inequality. Consequently, we achieve an O (^-2 | |^2) stabilization parameter, which guarantees global-in-time energy stability. The proposed framework is quite general and can be extended to other single-step schemes, such as implicit-explicit Runge–Kutta and exponential Runge–Kutta schemes, as well as to phase field models, including the classic Allen–Cahn and Cahn–Hilliard-type equations.