A new causal general relativistic formulation for dissipative continuum fluid and solid mechanics and its solution with high-order ADER schemes

TL;DR

This paper introduces a comprehensive, covariant general relativistic theory for dissipative and non-dissipative continuum mechanics, unifying fluids and solids, and demonstrates its numerical solution using high-order ADER schemes.

Contribution

It presents the first unified relativistic theory capable of modeling viscous fluids and solid deformations simultaneously, with a focus on hyperbolic PDEs and numerical methods.

Findings

Finite speed propagation of perturbations in the model

Well-posed Cauchy problem for arbitrary initial data

Successful numerical simulations in relativistic flows and solid deformations

Abstract

We present a unified causal general relativistic formulation of dissipative and non-dissipative continuum mechanics. The presented theory is the first general relativistic theory that can deal simultaneously with viscous fluids as well as irreversible deformations in solids and hence it also provides a fully covariant formulation of the Newtonian continuum mechanics in arbitrary curvilinear spacetimes. In such a formulation, the matter is considered as a Riemann-Cartan manifold with non-vanishing torsion and the main field of the theory being the non-holonomic basis tetrad field also called four-distortion field. Thanks to the variational nature of the governing equations, the theory is compatible with the variational structure of the Einstein field equations. Symmetric hyperbolic equations are the only admissible equations in our unified theory and thus, all perturbations propagate at finite speeds (even in the diffusive regime) and the Cauchy problem for the governing PDEs is locally well-posed for arbitrary and regular initial data which is very important for the numerical treatment of the presented model. Nevertheless, the numerical solution of the discussed hyperbolic equations is a challenging task because of the presence of the stiff algebraic source terms of relaxation type and non-conservative differential terms. Our numerical strategy is thus based on an advanced family of high-accuracy ADER Discontinuous Galerkin and Finite Volume methods which provides a very efficient framework for general relaxation hyperbolic PDE systems. An extensive range of numerical examples is presented demonstrating the applicability of our theory to relativistic flows of viscous fluids and deformation of solids in Minkowski and curved spacetimes.