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.

翻译：拉格朗日气体动力学方程属于双曲型与热力学相容（HTC）偏微分方程组这一大类，这类方程满足熵不等式（热力学第二定律）并守恒总能量（热力学第一定律）。本文旨在非结构化网格上构建一种新型热力学相容的单元中心拉格朗日有限体积格式。与现有格式不同，我们选择直接离散熵不等式，从而通过其他方程的新热力学相容离散方式自然导出总能量守恒。首先将控制方程改写为涨落形式，随后依据Abgrall等人近期提出的方法，利用网格节点上定义的标量修正因子对非相容中心数值通量进行修正。该方法完美契合单元中心拉格朗日有限体积法常用的节点求解器框架。我们设计了半离散熵守恒与熵稳定拉格朗日格式，并通过基于先验或后验间断解检测器的凸组合实现两者的恰当融合。严格证明了格式在能量范数下的非线性稳定性，并确保密度与压力的正性保持。此外，该格式在等熵流动中呈现零数值扩散且仍保持非线性稳定。通过拉格朗日水动力学经典算例验证了新格式的收敛性与鲁棒性，并定量对比了其与经典拉格朗日有限体积方法的数值耗散特征。