# Well-balanced finite volume schemes for hydrodynamic equations with general free energy

## TL;DR

This paper introduces well-balanced finite volume schemes for hydrodynamic equations with free energy, enabling precise stationary state preservation and stability analysis in complex systems like phase transitions and chemotaxis.

## Contribution

The paper develops novel finite volume schemes that preserve stationary states at machine precision for hydrodynamic systems with free energy, including nonlinear damping effects.

## Key findings

- Schemes accurately preserve stationary states at machine precision.
- Effective analysis of stability in phase transitions and chemotaxis models.
- Validated through extensive numerical tests.

## Abstract

Well balanced and free energy dissipative first- and second-order accurate finite volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The natural Liapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analog property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. These schemes allow for analysing accurately the stability properties of stationary states in challeging problems such as: phase transitions in collective behavior, generalized Euler-Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories; having done a careful validation in a battery of relevant test cases.