https://hal-ens.archives-ouvertes.fr/hal-03872221Couéraud, BenjaminBenjaminCouéraudGay-Balmaz, FrançoisFrançoisGay-BalmazENS-PSL - École normale supérieure - Paris - PSL - Université Paris sciences et lettresCNRS - Centre National de la Recherche ScientifiqueVariational discretization of the Navier-Stokes-Fourier systemHAL CCSD2022[MATH] Mathematics [math]Gay-Balmaz, François2022-11-25 15:30:452023-03-24 14:53:292022-11-25 15:30:45enPreprints, Working Papers, ...1This paper presents the variational discretization of the compressible Navier-Stokes-Fourier system, in which the viscosity and the heat conduction terms are handled within the variational approach to nonequilibrium thermodynamics as developed by one of the authors. In a first part, we review the variational framework for the Navier-Stokes-Fourier (NSF) system in the smooth setting. In a second part, we review a discrete exterior calculus based on discrete diffeomorphisms then proceed to establish the spatially discretized variational principle for the NSF system through the use of this discrete exterior calculus, which yields a semi-discrete nonholonomic variational principle, as well as semi-discrete evolution equations. In order to avoid important technical difficulties, further treatment of the phenomenological constraint is needed. In a third part we discretize in time the spatial variational principle underlying the NSF system by extending previous work of the authors, which at last yields a nonholonomic variational integrator for the NSF system, as well as fully discrete evolution equations.