An exponential supplementary variable method for ternary Cahn-Hilliard phase-field model

Developing efficient, structure-preserving numerical algorithms for multi-component phase-field models presents significant challenges, primarily due to the intricate nonlinear coupling of variables and the inherent stiffness of these models. In this study, we introduce an exponential supplementary variable method (ESVM) for solving the ternary Cahn-Hilliard phase field model. Our approach begins with the application of the supplementary variable approach (SVM) to reformulate the target model into an equivalent system, coupled with its original energy equation. We rigorously establish the theoretical equivalence between the system reconstructed via the supplementary variable method and the original model. In order to mitigate the stiffness and nonlinearity of the ternary Cahn-Hilliard phase-field model, we devise an efficient second-order algorithm that preserves the original energy dissipation law. This is achieved by synergizing the SVM method with the exponential integrating factor method and utilizing prediction-correction techniques within the framework of this new method. This class of ESVM methods not only facilitates straightforward extension to other models for the development of structure-preserving algorithms but also introduces a highly efficient solver that markedly reduces computational costs, thereby enabling the effective implementation of the proposed scheme. Numerical experiments conclusively demonstrate the superior overall performance of our newly developed scheme compared to existing methods.