We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure-preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. In this setting, discrete vector fields correspond to a certain subspace of the Lie algebra of this group. This subspace is shown to be isomorphic to a Raviart–Thomas finite element space. The resulting finite element discretization corresponds to a weak form of the compressible fluid equation that does not seem to have been used in the finite element literature. It extends previous work done on incompressible flows and at the lowest order on compressible flows. We illustrate the conservation properties of the scheme with some numerical simulations.

我们提出了可压缩流的有限元变分积分器。通过以保结构的方式离散微分群上流体动力学的李群公式和相关的变分原理，得出了数值方案。给定流体域上的三角剖分，将微分同构的离散组定义为有限元函数空间的线性同构组的某个子组。在这种设置下，离散矢量场对应于该组李代数的某个子空间。该子空间显示为与Raviart–Thomas有限元空间同构。所得的有限元离散化对应于可压缩流体方程的一种弱形式，似乎在有限元文献中并未使用。它扩展了先前在不可压缩流上所做的工作，并以最低顺序扩展了可压缩流。我们通过一些数值模拟说明了该方案的守恒性质。