Unconditionally stable methods for simulating multi-component two-phase interface models with Peng–Robinson equation of state and various boundary conditions

In this paper, we consider multi-component dynamic two-phase interface models, which are formulated by the Cahn–Hilliard system with Peng–Robinson equation of state and various boundary conditions. These models can be derived from the minimum problems of Helmholtz free energy or grand potential in the realistic thermodynamic systems. The resulted Cahn–Hilliard systems with various boundary conditions are fully coupled and strongly nonlinear. A linear transformation is introduced to decouple the relations between different components, and as a result, the models are simplified. From this, we further propose a semi-implicit unconditionally stable time discretization scheme, which allows us to solve the Cahn–Hilliard system by a decoupled way, and thus, our method can significantly reduce the computational cost and memory requirements. The mixed finite element methods are employed for the spatial discretization, and the approximate errors are also analyzed for both space and time. Numerical examples are tested to demonstrate the efficiency of our proposed methods.