Linear and Unconditionally Energy Stable Schemes for the Multi-Component Two-Phase Diffuse Interface Model with Peng-Robinson Equation of State

In this paper we consider numerical solutions of the diffuse interface model with Peng-Robinson equation of state for the multi-component two-phase fluid system, which describes real states of hydrocarbon fluids in petroleum industry. A major challenge is to develop appropriate temporal discretizations to overcome the strong nonlinearity of the source term and preserve the energy dissipation law in the discrete sense. Efficient first and second order time stepping schemes are designed based on the "Invariant Energy Quadratization" approach and the stabilized method. The resulting temporal semi-discretizations by both schemes lead to linear systems that are symmetric and positive definite at each time step, and their unconditional energy stabilities are rigorously proven. Numerical experiments are presented to demonstrate accuracy and stability of the proposed schemes.