  In this paper we present a new family of semi-discrete and fully-discrete
finite volume schemes for overdetermined, hyperbolic and thermodynamically
compatible PDE systems. In the following we will denote these methods as HTC
schemes. In particular, we consider the Euler equations of compressible
gasdynamics, as well as the more complex Godunov-Peshkov-Romenski (GPR) model
of continuum mechanics, which, at the aid of suitable relaxation source terms,
is able to describe nonlinear elasto-plastic solids at large deformations as
well as viscous fluids as two special cases of a more general first order
hyperbolic model of continuum mechanics. The main novelty of the schemes
presented in this paper lies in the fact that we solve the \textit{entropy
inequality} as a primary evolution equation rather than the usual total energy
conservation law. Instead, total energy conservation is achieved as a mere
consequence of a thermodynamically compatible discretization of all the other
equations. For this, we first construct a discrete framework for the
compressible Euler equations that mimics the continuous framework of Godunov's
seminal paper \textit{An interesting class of quasilinear systems} of 1961
\textit{exactly} at the discrete level. All other terms in the governing
equations of the more general GPR model, including non-conservative products,
are judiciously discretized in order to achieve discrete thermodynamic
compatibility, with the exact conservation of total energy density as a direct
consequence of all the other equations. As a result, the HTC schemes proposed
in this paper are provably marginally stable in the energy norm and satisfy a
discrete entropy inequality by construction. We show some computational results
obtained with HTC schemes in one and two space dimensions, considering both the
fluid limit as well as the solid limit of the governing partial differential
equations.
