Convergent numerical method for the Navier-Stokes-Fourier system: a stabilized scheme

0509632 - MÚ 2020 RIV GB eng J - Journal Article
Hošek, Radim - She, Bangwei
Convergent numerical method for the Navier-Stokes-Fourier system: a stabilized scheme.
IMA Journal of Numerical Analysis. Roč. 39, č. 4 (2019), s. 2045-2068. ISSN 0272-4979. E-ISSN 1464-3642
EU Projects: European Commission(XE) 320078 - MATHEF
Institutional support: RVO:67985840
Keywords : Navier-Stokes-Fourier system
OECD category: Pure mathematics
Impact factor: 2.275, year: 2019
Method of publishing: Limited access
http://dx.doi.org/10.1093/imanum/dry057

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.
Permanent Link: http://hdl.handle.net/11104/0300314