标题： 一种适用于拉格朗日气体动力学的新的热力学相容有限体积格式 显示英文标题

标题： A new thermodynamically compatible finite volume scheme for Lagrangian gas dynamics

摘要： 拉格朗日气体动力学的方程属于超定双曲和热力学相容（HTC）偏微分方程的大类。 它们满足熵不等式（热力学第二定律）并守恒总能量（热力学第一定律）。 本工作的目标是在非结构化网格上构建一种新颖的热力学相容的单元中心拉格朗日有限体积方案。 与现有方案不同，我们选择直接离散熵不等式，因此通过其他方程的新热力学相容离散化得到总能量守恒。 首先，控制方程以波动形式写出。 接下来，根据Abgrall等人最近引入的方法对非相容的中心数值通量进行修正，该方法使用在网格节点上定义的标量修正因子。 这完美地融入了通常在单元中心拉格朗日有限体积方法中采用的节点求解器的形式。 设计了半离散的熵守恒和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适当结合在一起。 严格证明了能量范数下的非线性稳定性，新方案对于密度和压力具有可证明的正性保持性。 此外，它们在等熵流动中表现出零数值扩散，同时仍保持非线性稳定性。 新方案针对拉格朗日流体力学的经典基准进行测试，评估其收敛性和鲁棒性，并将其数值耗散与经典拉格朗日有限体积方法进行比较。 显示英文摘要

摘要： 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.