A Thermodynamically Consistent High-Order Framework for Staggered Lagrangian Hydrodynamics

Abstract

We present a consistent high-order staggered Lagrangian hydrodynamics framework designed to reconcile an underlying disparity in existing curvilinear formulations: the mismatch between quadrature-based "strong" mass conservation and the discrete degrees of freedom (DOFs) of thermodynamic variables. By mathematically coupling the numerical quadrature rule with the density representation, our approach ensures rigorous point-wise consistency between density, internal energy, and pressure. This synchronization eliminates the ambiguity of equation-of-state (EOS) updates inherent in previous high-order staggered methods. To stabilize the discretization, we develop a high-order generalization of the subzonal pressure method by conceptually enriching the pressure field from the $Q^{m-1}$ to the $Q^m$ finite element space. We prove that evaluating this enriched field using a high-order quadrature rule naturally generates a restorative anti-hourglass force, which exactly recovers the classical $Q^1-P^0$ compatible hydrodynamics algorithm as a limiting case for $m=1$. Furthermore, we introduce a concise, algorithmic formulation of tensor artificial viscosity that streamlines implementation and significantly reduces computational overhead in high-order settings. The resulting framework yields strictly diagonal mass matrices for both momentum and energy equations, enabling highly efficient, fully explicit time integration without global linear solves. Extensive numerical benchmarks, including smooth convergence tests and complex shock-dominated flows, demonstrate that the proposed method achieves optimal high-order accuracy while maintaining superior geometric robustness.