A Conservative Finite Element Method for the Incompressible Euler Equations with Variable Density
A Conservative Finite Element Method for the Incompressible Euler Equations with Variable Density
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DOI：
10.1016/j.jcp.2020.109439
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发表时间：
2019-10
期刊：
影响因子：
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通讯作者：
Evan S. Gawlik;F. Gay‐Balmaz
中科院分区：
文献类型：
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作者：
Evan S. Gawlik;F. Gay‐Balmaz
我们构造了变密度不可压缩欧拉方程的有限元离散和时间步进格式，该格式精确地保持了总质量、总平方密度、总能量和逐点不可压缩。该方法用Raviart-Thomas或Brezzi-Douglas-Marini有限元逼近速度，用间断多项式逼近密度和压力。为了实现上述守恒量的精确保持，我们利用了一个很少使用的动量方程的弱公式和一个与中点规则相似但不完全相同的二阶时间步长格式。我们还描述并证明了该方法的迎风版本的稳定性。我们给出的数值例子证明了该方法的收敛阶。
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.