Machine-precision energy conservative reduced models for Lagrangian hydrodynamics by quadrature methods

We present an energy conservative, quadrature based model reduction framework for the compressible Euler equations of Lagrangian hydrodynamics. Building on a finite element discretization of the governing equations, we develop reduced models using data based reduced basis functions and the empirical quadrature procedure (EQP). We introduce a strongly energy conservative variant of EQP that enforces exact energy conservation in the reduction process. Numerical experiments for four benchmark problems – Sedov blast, Gresho vortex, triple point and Taylor-Green vortex – demonstrate that the numerical implementation of our proposed method conserves total energy to near machine precision, while maintaining accuracy comparable to the basic EQP formulation.