A new continuum model for general relativistic viscous heat-conducting media

Abstract

<jats:p>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 viscosity, heat conductivity, etc., 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.</jats:p> <jats:p>This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.</jats:p>

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

“…An extension of this class of models to compressible multi-phase flows was forwarded in [103,104,106]. The SHTC framework remains valid even in the context of special and general relativity, see [76,105]. Recently, a connection between the class of symmetric hyperbolic and thermodynamically compatible systems and Hamiltonian mechanics was rigorously established in [92].…”

A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

“…Future work will consider the extension of P N P M scheme with N > 0, M > N to unstructured moving meshes [18,63], in particular for regenerating Voronoi tessellations [59,60], and to the three-dimensional case. Finally, due to their low memory consumption and their gain in computational efficiency compared to DG schemes, they will be also considered for astrophysical applications [42,44,61] and the unified first order hyperbolic model of continuum mechanics proposed in [25,48,49,93,98], where a large number of conserved variables has to be discretized. Due to their accuracy and compact stencil in the future we also plan to use P N P M schemes with a posteriori subcell finite volume limiter in the context of hyperbolic reformulations of nonlinear dispersive systems and wave propagation problems, see e.g.…”

Nonequilibrium thermodynamics: emergent and fundamental

“…Dissipative processes tend to equilibrium; entropy is a kind of Lyapunov functional. This aspect is crucial in [1][2][3], important but less apparent in [4,5]. -Symmetric hyperbolic evolution.…”

“…This property is somehow connected to ideal systems, without dissipation, but a thermodynamic framework provides the natural variables and the necessary potential formulation. Symmetric hyperbolicity is a basic requirement in [2,4,6] but plays an important role in [7,8], too. -Space-time embedding.…”

“…Dissipation does not depend on the observer. This requirement is evident in the relativistic treatment of [4], and present also in [2,5,9]. Also, the preferred application areas influence modelling methods.…”