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<p>Surface-tension-driven interfacial evolution provides a representative class of stiff continuum problems. In the present work, stiffness is understood in the numerical-analysis sense: after spatial discretization, the governing equations give rise to a semi-discrete evolution system containing modes with strongly separated decay rates. In this work, we develop a unified semi-implicit second-order time-integration framework for stiff phase-field systems. Within a finite-element weak formulation, an implicit–explicit (IMEX) strategy is employed, in which stiff linear interfacial terms are treated implicitly while nonlinear contributions are evaluated explicitly, leading to a linear-per-step update structure. A standard semi-implicit BDF2 scheme is first constructed and systematically assessed, demonstrating significantly reduced temporal dissipation compared to first-order schemes and enabling stable time steps nearly an order of magnitude larger in stiffness-dominated regimes. To further enhance robustness under extreme transient stiffness, a generalized BDF2 formulation with controlled numerical dissipation is introduced. Its second-order consistency is rigorously established at a shifted time node, providing a theoretical foundation for stability enhancement and for a consistent extension to variable time-step integration. Building on this formulation, a free-energy-rate-based adaptive time-stepping strategy is proposed to dynamically regulate temporal resolution according to the evolving free-energy dissipation behavior associated with stiff transients. Numerical studies based on surface-tension-driven viscous deformation benchmarks demonstrate that the proposed adaptive BDF2 framework substantially reduces computational cost while preserving accuracy, monotonic energy dissipation, and thermodynamic consistency. The developed methodology provides a robust and efficient numerical strategy for stiff phase-field simulations and is expected to be applicable to a broad class of interfacial evolution problems governed by strong temporal stiffness.</p>

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A semi-implicit BDF2 framework with generalized stabilization and energy-rate-based adaptive time integration for stiff phase-field systems

  • 期刊论文

  • 发表日期：

    2026年6月20日

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  • Chao He,
  • Bo Qian,
  • Qingcheng Yang,
  • Jiwei Zhang,
  • Chengchao Zhao

摘要

Surface-tension-driven interfacial evolution provides a representative class of stiff continuum problems. In the present work, stiffness is understood in the numerical-analysis sense: after spatial discretization, the governing equations give rise to a semi-discrete evolution system containing modes with strongly separated decay rates. In this work, we develop a unified semi-implicit second-order time-integration framework for stiff phase-field systems. Within a finite-element weak formulation, an implicit–explicit (IMEX) strategy is employed, in which stiff linear interfacial terms are treated implicitly while nonlinear contributions are evaluated explicitly, leading to a linear-per-step update structure. A standard semi-implicit BDF2 scheme is first constructed and systematically assessed, demonstrating significantly reduced temporal dissipation compared to first-order schemes and enabling stable time steps nearly an order of magnitude larger in stiffness-dominated regimes. To further enhance robustness under extreme transient stiffness, a generalized BDF2 formulation with controlled numerical dissipation is introduced. Its second-order consistency is rigorously established at a shifted time node, providing a theoretical foundation for stability enhancement and for a consistent extension to variable time-step integration. Building on this formulation, a free-energy-rate-based adaptive time-stepping strategy is proposed to dynamically regulate temporal resolution according to the evolving free-energy dissipation behavior associated with stiff transients. Numerical studies based on surface-tension-driven viscous deformation benchmarks demonstrate that the proposed adaptive BDF2 framework substantially reduces computational cost while preserving accuracy, monotonic energy dissipation, and thermodynamic consistency. The developed methodology provides a robust and efficient numerical strategy for stiff phase-field simulations and is expected to be applicable to a broad class of interfacial evolution problems governed by strong temporal stiffness.