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This paper is concerned with the numerical solution of the unified firstorder hyperbolic formulation of continuum mechanics recently proposed byPeshkov & Romenski, denoted as HPR model. In that framework, the viscousstresses are computed from the so-called distortion tensor A, which is one ofthe primary state variables. A very important key feature of the model is itsability to describe at the same time the behavior of inviscid and viscouscompressible Newtonian and non-Newtonian fluids with heat conduction, as wellas the behavior of elastic and visco-plastic solids. This is achieved via astiff source term that accounts for strain relaxation in the evolutionequations of A. Also heat conduction is included via a first order hyperbolicevolution equation of the thermal impulse, from which the heat flux iscomputed. The governing PDE system is hyperbolic and fully consistent with theprinciples of thermodynamics. It is also fundamentally different from firstorder Maxwell-Cattaneo-type relaxation models based on extended irreversiblethermodynamics. The connection between the HPR model and the classicalhyperbolic-parabolic Navier-Stokes-Fourier theory is established via a formalasymptotic analysis in the stiff relaxation limit. From a numerical point ofview, the governing partial differential equations are very challenging, sincethey form a large nonlinear hyperbolic PDE system that includes stiff sourceterms and non-conservative products. We apply the successful family of one-stepADER-WENO finite volume and ADER discontinuous Galerkin finite element schemesin the stiff relaxation limit, and compare the numerical results with exact ornumerical reference solutions obtained for the Euler and Navier-Stokesequations. To show the universality of the model, the paper is rounded-off withan application to wave propagation in elastic solids.

High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids