We propose a combined finite volume--finite element method for the compressible Navier–Stokes–Fourier system. A finite volume approximation is used for the density and energy equations while a finite element discretization based on the nonconforming Crouzeix–Raviart element is applied to the momentum equation. We show the stability, the consistency and finally the convergence of the scheme (up to a subsequence) toward a suitable weak solution. We are interested in the diffusive term in the form of divergence of the symmetric velocity gradient instead of the classical Laplace form appearing in the momentum equation. As a consequence, there emerges the need to add a stabilization term that substitutes the role of Korn’s inequality which does not hold in the Crouzeix–Raviart element space. The present work is a continuation of Feireisl, E., Hošek, R. & Michálek, M. (2016, A convergent numerical method for the Navier–Stokes–Fourier system. IMA J. Numer. Anal., 36, 1477--1535), where a similar scheme is studied for the case of classical Laplace diffusion. We compare the two schemes and point out that the discretization of the energy diffusion terms in the reference scheme is not compatible with the model. Finally, we provide several numerical experiments for both schemes to demonstrate the numerical convergence, positivity of the discrete density, as well as the difference between the schemes.

中文翻译：

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Navier–Stokes–Fourier系统的收敛数值方法：稳定方案

我们为可压缩的Navier–Stokes–Fourier系统提出了一种有限体积-有限元组合方法。密度和能量方程式使用有限体积近似，而动量方程式中使用基于非一致性Crouzeix-Raviart元素的有限元离散化。我们展示了该方案的稳定性，一致性以及最终收敛（直到一个子序列）朝向一个合适的弱解。我们对对称速度梯度的发散形式而不是在动量方程中出现的经典拉普拉斯形式的扩散项感兴趣。结果，出现了需要添加一个稳定项来代替在Crouzeix-Raviart元素空间中不成立的Korn不等式的作用。本工作是Feireisl，E.，Hošek，R.的继续。IMA J. Numer。肛门 ，第36卷，第1477--1535页），其中针对经典拉普拉斯扩散的情况研究了类似的方案。我们比较了两种方案，并指出参考方案中能量扩散项的离散化与模型不兼容。最后，我们提供两种方案的数值实验，以证明数值收敛性，离散密度的正性以及方案之间的差异。