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

Published on by Sergio Perez
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.

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We are indebted to P. Yatsyshin and M. A. Durán-Olivencia from the Chemical Engineering Department of Imperial College (IC) for numerous stimulating discussions on statistical mechanics of classical fluids and density functional theory. J. A. Carrillo was partially supported by EPSRC via Grant Number EP/P031587/1 and acknowledges support of the IBM Visiting Professorship of Applied Mathematics at Brown University. S. Kalliadasis was partially supported by EPSRC via Grant Number EP/L020564/1. S. P. Perez acknowledges financial support from the IC President’s PhD Scholarship and thanks Brown University for hospitality during a visit in April 2018. C.-W. Shu was partially supported by NSF via Grant Number DMS-1719410.