Operator splitting decoupled energy-stable numerical schemes for phase field grain growth models

Abstract

Phase-field methods have been widely employed to study grain growth in materials science. These grain-growth phase-field models are typically derived from variational principles, which ensure thermodynamic consistency and energy dissipation properties. Numerical discretizations that inherit these properties are known as energy-stable or structure-preserving schemes. However, existing energy-stable schemes for phase-field grain-growth models typically require solving fully coupled systems, which are computationally expensive, especially when the number of grains is large. In this paper, we develop a class of computationally efficient, fully decoupled numerical schemes based on operator splitting techniques. We present this decoupling idea by integrating it with several existing energy-stable strategies, including Energy Quadratization (EQ), Relaxed Energy Quadratization (REQ), the stabilization method, and the convex splitting method. The resulting schemes are unconditionally energy-stable, i.e., they satisfy a discrete energy dissipation law without restrictions on the time-step sizes. The proposed schemes are efficient because they only require updating each grain through small, independent subproblems at each time step. We present first and second-order formulations and provide rigorous energy analyses. Numerical convergence tests and benchmark examples are presented to validate the accuracy, efficiency, and robustness of our proposed decoupled numerical methods for simulating large grain systems.