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

The development of well-balanced numerical methods able to discretely preserve steady states of balance laws has attracted considerable attention since the first works nearly twenty years ago. Standard finite-volume approaches fail to accurately resolve the steady states from balance laws, in which the fluxes need to be exactly balanced with the source terms. To correct this deficiency, the well-balanced schemes are designed so as to discretely satisfy this balance when the steady state is reached. But well-balanced schemes for hydrodynamic equations with a general free energy have not been developed as of yet.

Here we outline a well-balanced finite volume scheme for a general free energy, which could contain different dependencies with respect to the density and external or interaction potentials. This scheme is sufficiently flexible and can be applied to applications involving shallow water equations, cell chemotaxis, and dynamic-density functional theory, to name but a few. We show that the first and second-order schemes preserve the steady states and satisfy natural properties of the original system.