Flux Globalization Based Well-Balanced Central-Upwind Schemes for Hydrodynamic Equations with General Free Energy

Abstract

AbstractWe develop flux globalization based well-balanced central-upwind schemes for hydrodynamic equations with general free energy. The proposed schemes are well-balanced in the sense that they are capable of exactly preserving quite complicated steady-state solutions and also exactly capturing traveling waves, even when vacuum regions are present. In order to accurately track interfaces of the vacuum regions and near vacuum parts of the solution, we use the technique introduced in Chertock et al. (J Sci Comput 90:Paper No. 9, 2022) and design a hybrid approach: inside the no vacuum regions, we use the flux globalization based well-balanced central-upwind scheme, while elsewhere we implement the central-upwind scheme similar to the one proposed in Bollermann et al. (J Sci Comput 56:267–290, 2013) in the context of wet/dry fronts in the shallow water equations. The advantages of the proposed schemes are demonstrated on a number of challenging numerical examples.