Highly efficient variant of SAV approach for the incompressible multi-component phase-field fluid models

Here, we study the multi-component Cahn-Hilliard (multiCH) system and the three-phase fluid system by a novel numerical scheme. The nonlinear terms in the multiCH system are obstructors to establishing numerical schemes. In our proposed method, several scalar auxiliary variables are defined to represent the nonlinear terms. Then, the control equations can be converted into equivalent forms where the nonlinear terms are replaced by the scalar auxiliary variables. Finally, we only need to discretize the equivalence equation, where the second order backward difference formula (BDF2) and the second-order central difference method are applied in the time and the spatial domain, respectively. The entire calculation process is linear and decoupled. In each time iteration, we apply a fast linear multigrid algorithm to the constant coefficient linear elliptic equation. For the incompressible flow part, we employ a pressure correction method to decouple velocity and pressure during time discretization. We rigorously prove that the numerical solution is unique and shows that the resulting numerical solution satisfies the dissipation of modified energy. The temporal second-order accuracy test, the energy dissipation test, the comparison test with the classical SAV in CPU time, and other various numerical tests show the good performance of our method.