  This 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.
