  This paper is concerned with the numerical solution of the unified first
order hyperbolic formulation of continuum mechanics recently proposed by
Peshkov & Romenski, denoted as HPR model. In that framework, the viscous
stresses are computed from the so-called distortion tensor A, which is one of
the primary state variables. A very important key feature of the model is its
ability to describe at the same time the behavior of inviscid and viscous
compressible Newtonian and non-Newtonian fluids with heat conduction, as well
as the behavior of elastic and visco-plastic solids. This is achieved via a
stiff source term that accounts for strain relaxation in the evolution
equations of A. Also heat conduction is included via a first order hyperbolic
evolution equation of the thermal impulse, from which the heat flux is
computed. The governing PDE system is hyperbolic and fully consistent with the
principles of thermodynamics. It is also fundamentally different from first
order Maxwell-Cattaneo-type relaxation models based on extended irreversible
thermodynamics. The connection between the HPR model and the classical
hyperbolic-parabolic Navier-Stokes-Fourier theory is established via a formal
asymptotic analysis in the stiff relaxation limit. From a numerical point of
view, the governing partial differential equations are very challenging, since
they form a large nonlinear hyperbolic PDE system that includes stiff source
terms and non-conservative products. We apply the successful family of one-step
ADER-WENO finite volume and ADER discontinuous Galerkin finite element schemes
in the stiff relaxation limit, and compare the numerical results with exact or
numerical reference solutions obtained for the Euler and Navier-Stokes
equations. To show the universality of the model, the paper is rounded-off with
an application to wave propagation in elastic solids.
