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Well balanced and free energy dissipative first- and second-order accuratefinite volume schemes are proposed for a general class of hydrodynamic systemswith linear and nonlinear damping. The natural Liapunov functional of thesystem, given by its free energy, allows for a characterization of thestationary states by its variation. An analog property at the discrete levelenables us to preserve stationary states at machine precision while keeping thedissipation of the discrete free energy. These schemes allow for analysingaccurately the stability properties of stationary states in challeging problemssuch as: phase transitions in collective behavior, generalized Euler-Poissonsystems in chemotaxis and astrophysics, and models in dynamic densityfunctional theories; having done a careful validation in a battery of relevanttest cases.

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