Consistency-enhanced SAV BDF2 time-marching method with relaxation for the incompressible Cahn–Hilliard–Navier–Stokes binary fluid model

In recent years, the scalar auxiliary variable (SAV) approach is popular to design decoupled, linear, and energy dissipation-preserving scheme for phase-field models. By introducing an appropriate time-dependent auxiliary variable, the original equations are transformed into equivalent forms and the nonlinear and coupling terms can be easily handled in an explicit manner. The estimation of energy stability is easy to prove and the numerical implementation is efficient. By utilizing the “zero-energy-contribution” property and defining another time-dependent auxiliary variable, the SAV approach can be extended to develop linear and energy-stable schemes for various phase-field fluid models. However, the discrete version of energy stability resulting from SAV approach corresponds to a modified energy instead of the original one. Generally, the discrete modified energy and discrete original energy are not consistent. Moreover, the discrete auxiliary variable related to “zero-energy-contribution” property might be not consistent with its continuous value 1. To enhance the consistency, we herein develop a simple and practical relaxation technique for the SAV BDF2 scheme of Cahn–Hilliard–Navier–Stokes model. The relaxation technique does not obviously increase the computational costs and still leads to provable energy dissipation law. In each time step, the calculation is efficient because we only need to solve several elliptic equations with constant coefficients. Numerical experiments indicate that the proposed method has desired accuracy, energy stability, and capability for incompressible two-phase fluid flows.