A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.