Consistently energy-stable decoupled method with second-order accuracy and lower density bounds for the incompressible fluid flows with variable density

In this paper, we develop an energy-stable and linear numerical scheme for solving the incompressible fluid flows with variable density. To facilitate the construction of numerical method, two time-dependent auxiliary variables are introduced to recast the original governing equations into equivalent forms. Based on the equivalent equations, we present a semi-implicit time-marching scheme with second-order backward differentiation formula (BDF2), where the linear term and auxiliary variables are implicitly treated. Using a splitting technique, the proposed scheme can be easily solved in a totally decoupled manner. We analytically estimate the energy stability and the preservation of lower density bounds. In each time step, two simple correction steps are used to improve the energy consistency. Several numerical experiments are implemented to validate the accuracy, energy stability, and lower density bounds of the proposed method. Moreover, the Rayleigh–Taylor instability and incompressible flows with variable viscosity are simulated to further show the good capabilities.