We construct a finite element discretization and time-stepping scheme for the incompressible Euler equations with variable density that exactly preserves total mass, total squared density, total energy, and pointwise incompressibility. The method uses Raviart-Thomas or Brezzi-Douglas-Marini finite elements to approximate the velocity and discontinuous polynomials to approximate the density and pressure. To achieve exact preservation of the aforementioned conserved quantities, we exploit a seldom-used weak formulation of the momentum equation and a second-order time-stepping scheme that is similar, but not identical, to the midpoint rule. We also describe and prove stability of an upwinded version of the method. We present numerical examples that demonstrate the order of convergence of the method.

我们为可变密度的不可压缩的Euler方程构造了一个有限元离散化和时间步进方案，该方程精确地保留了总质量，总平方密度，总能量和逐点不可压缩性。该方法使用Raviart-Thomas或Brezzi-Douglas-Marini有限元来近似速度，使用不连续多项式来近似密度和压力。为了精确地保存上述守恒量，我们利用了动量方程的很少使用的弱公式和类似于中点法则但不相同的二阶时间步长方案。我们还将描述并证明该方法上风版本的稳定性。我们提供了数值示例，说明了该方法收敛的顺序。