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

## TL;DR

This paper introduces an energy conservative reduced modeling approach for Lagrangian hydrodynamics that ensures near machine-precision energy conservation while maintaining high accuracy, using a quadrature-based model reduction framework.

## Contribution

The paper develops a strongly energy conservative variant of empirical quadrature for reduced models of Euler equations, ensuring exact energy conservation during model reduction.

## Key findings

- Conserves total energy to near machine precision.
- Maintains accuracy comparable to basic EQP methods.
- Validated on four benchmark problems.

## Abstract

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.