A boosted second-order convex splitting algorithm based on gradient flows

TL;DR

This paper proposes a second-order convex splitting scheme for gradient flows in phase-field models, combining BDF2 and Adams-Bashforth methods, with proven convergence and improved efficiency over existing methods.

Contribution

It introduces a novel second-order scheme with rigorous convergence analysis and practical efficiency enhancements for gradient flow simulations.

Findings

The scheme is energy stable and convergent under mild conditions.

Numerical experiments show improved computational efficiency.

The method is robust for smooth and nonsmooth energy functionals.

Abstract

This paper introduces a second-order convex splitting scheme for gradient flows arising in phase-field models, based on the backward differentiation formula (BDF2) for the implicit part and the Adams-Bashforth method for the nonlinear and explicit component. The method is formulated and analyzed in finite-dimensional spaces, where energy stability plays a central role in establishing rigorous convergence properties. By leveraging the Kurdyka-\L ojasiewicz framework, we prove the global convergence of the discrete trajectories generated by the scheme, even in the presence of nonsmooth energy functionals, under mild assumptions on the time-step size. The Armijo line search strategy and the classical preconditioning strategies, such as symmetric Gauss-Seidel and Jacobi, are incorporated to improve its computational efficiency. Numerical experiments confirm that the proposed method achieves computational efficiency compared to existing first-order splitting approaches and other accelerated splitting algorithms, while maintaining robustness in both smooth and nonsmooth regimes.