  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.
