A unified energy-stable time integration framework including adaptive time-stepping for phase-field modeling

This paper develops a unified second-order accurate, and fully implicit time integration framework based on the Generalized Single-Step Single-Solve architecture to solve the Allen–Cahn and Cahn–Hilliard equations. The proposed framework encompasses the non-dissipative Crank-Nicolson scheme and numerous dissipative variants, such as Generalized- α and BDF2, featuring flexibly controllable high-frequency numerical dissipation. By directly resolving the original nonlinear equations, we provide comprehensive analytical and numerical comparisons of various nonlinear discretization strategies regarding the energy stability and computational efficiency. Theoretical analysis proves that the framework preserves unconditional energy stability for the Crank-Nicolson scheme and establishes conditional stability bounds for the dissipative schemes. Furthermore, unlike traditional local-truncation-error estimators in the literature, we introduce a novel physical discrepancy indicator to drive the adaptive time-stepping by evaluating the intrinsic deviation between the nonlinear approximation and thermodynamic energy decay. This indicator is universally applicable across the entire family of proposed algorithms, ensuring robust convergence for various nonlinear discretizations once a single user-defined tolerance is met. Finally, two- and three-dimensional numerical examples rigorously verify the energy stability and accuracy, demonstrating the framework’s capability in capturing complex phase-field evolutions.