Numerical schemes for a unified first order hyperbolic system of continuum mechanics

In the first part of this talk we present the unified first order hyperbolic formulation of continuum mechanics of Godunov, Peshkov and Romenski (GPR). The PDE system belongs to the class of symmetric hyperbolic and thermodynamically compatible systems (SHTC), discovered by Godunov in 1961. The GPR model is a geometric approach to mechanics that is able to describe nonlinear elastoplastic solids and viscous Newtonian and non-Newtonian fluids within one and the same governing PDE system. This is achieved via appropriate relaxation source terms in the evolution equations. It can be shown that the GPR model reduces to the compressible Navier-Stokes equations in the stiff relaxation limit. The system is also able to describe material failure, crack generation and fatigue. In the absence of source terms, the GPR model is endowed with involutions, namely the distortion field A and the thermal impulse J need to remain curl-free. In the second part of the talk a family of high order ADER discontinuous Galerkin finite element schemes with a posteriori subcell finite volume limiter is introduced and applied to the GPR model. In the third part of the talk we present a new structure-preserving semi-implicit scheme that is able to preserve the curl-free property of both fields exactly also on the discrete level. The scheme is asymptotic-preserving in the stiff relaxation limit, where the scheme tends to an appropriate discretization of the compressible Navier-Stokes equations. In the final part of the talk we present a new thermodynamically compatible finite volume scheme that is exactly compatible with the overdetermined structure of the model at the semi-discrete level, making use of a discrete form of the continuous formalism introduced by Godunov in 1961. A very particular feature of our new thermodynamically compatible FV scheme is that it directly discretizes the entropy inequality, rather than the total energy conservation law. Energy conservation is instead achieved as a mere consequence of the scheme, thanks to the thermodynamically compatible discretization of all the other equations.