  The lack of formulation of macroscopic equations for irreversible dynamics of
viscous heat-conducting media compatible with the causality principle of
Einstein's Special Relativity and the Euler-Lagrange structure of General
Relativity is a long-lasting problem. In this paper, we propose a possible
solution to this problem in the framework of SHTC equations. The approach does
not rely on postulates of equilibrium irreversible thermodynamics but treats
irreversible processes from the non-equilibrium point of view. Thus, each
transfer process is characterized by a characteristic velocity of perturbation
propagation in the non-equilibrium state, as well as by an intrinsic
time/length scale of the dissipative dynamics. The resulting system of
governing equations is formulated as a first-order system of hyperbolic
equations with relaxation-type irreversible terms. Via a formal asymptotic
analysis, we demonstrate that classical transport coefficients such as the
viscosity and heat conductivity are recovered in leading terms of our theory as
effective transport coefficients. Some numerical examples are presented in
order to demonstrate the viability of the approach.
