Well-Balancing via Flux Globalization: Applications to Shallow Water Equations with Wet/Dry Fronts

TLDR: In this article, Cheng et al. study the flux globalization based central-upwind scheme for the Saint-Venant system of shallow water equations and develop a well-balanced scheme, which can accurately handle both still and moving-water equilibria.

Abstract: We study the flux globalization based central-upwind scheme from Cheng et al. (J Sci Comput 80:538–554, 2019) for the Saint-Venant system of shallow water equations. We first show that while the scheme is capable of preserving moving-water equilibria, it fails to preserve much simpler “lake-at-rest” steady states. We then modify the computation of the global flux variable and develop a well-balanced scheme, which can accurately handle both still- and moving-water equilibria. In addition, we extend the flux globalization based central-upwind scheme to the case when “dry” and/or “almost dry” areas are present. To this end, we introduce a hybrid approach: we use the flux globalization based scheme inside the “wet” areas only, while elsewhere we apply the central-upwind scheme from Bollermann et al. (J Sci Comput 56:267–290, 2013), which is designed to accurately capture wet/dry fronts. We illustrate the performance of the proposed schemes on a number of numerical examples.