For numerical schemes to the incompressible Navier-Stokes equations with variable density, it is a critical property to preserve the bounds of density. A bound-preserving high order accurate scheme can be constructed by using high order discontinuous Galerkin (DG) methods or finite volume methods with a bound-preserving limiter for the density evolution equation, with any popular numerical method for the momentum evolution. In this paper, we consider a combination of a continuous finite element method for momentum evolution and a bound-preserving DG method for density evolution. Fully explicit and explicit-implicit strong stability preserving Runge-Kutta methods can be used for the time discretization for the sake of bound-preserving. Numerical tests on representative examples are shown to demonstrate the performance of the proposed scheme.

对于具有可变密度的不可压缩的Navier-Stokes方程的数值方案，保持密度边界是至关重要的。可以通过使用高阶不连续Galerkin（DG）方法或有限体积方法（对于密度演化方程）使用保界限制器，以及任何流行的数值方法来构造动量演化，来构造保界高阶精确方案。在本文中，我们考虑了用于动量演化的连续有限元方法和用于密度演化的保界DG方法的组合。为了保留边界，可以将完全显式和显式隐式强稳定性保持Runge-Kutta方法用于时间离散化。通过对代表性示例进行数值测试，可以证明所提出方案的性能。