  The equations of Lagrangian gas dynamics fall into the larger class of
overdetermined hyperbolic and thermodynamically compatible (HTC) systems of
partial differential equations. They satisfy an entropy inequality (second
principle of thermodynamics) and conserve total energy (first principle of
thermodynamics). The aim of this work is to construct a novel thermodynamically
compatible cell-centered Lagrangian finite volume scheme on unstructured
meshes. Unlike in existing schemes, we choose to directly discretize the
entropy inequality, hence obtaining total energy conservation as a consequence
of the new thermodynamically compatible discretization of the other equations.
First, the governing equations are written in fluctuation form. Next, the
non-compatible centered numerical fluxes are corrected according to the
approach recently introduced by Abgrall et al., using a scalar correction
factor that is defined at the nodes of the grid. This perfectly fits into the
formalism of nodal solvers which is typically adopted in cell-centered
Lagrangian finite volume methods. Semi-discrete entropy conservative and
entropy stable Lagrangian schemes are devised, and they are adequately blended
together via a convex combination based on either a priori or a posteriori
detectors of discontinuous solutions. The nonlinear stability in the energy
norm is rigorously demonstrated and the new schemes are provably positivity
preserving for density and pressure. Furthermore, they exhibit zero numerical
diffusion for isentropic flows while being still nonlinearly stable. The new
schemes are tested against classical benchmarks for Lagrangian hydrodynamics,
assessing their convergence and robustness and comparing their numerical
dissipation with classical Lagrangian finite volume methods.
