Generalized Boltzmann equation for shallow water flows

Abstract

This invited review paper introduces the Boltzmann-based approach for the numerical modelling of surface water flows to hydroinformaticians. The paper draws upon earlier work by our group as well as others. This review formulates the generalized Boltzmann equation for 1D and 2D shallow water flows and shows that the statistical moments of these generalized equations provide the classical continuity and momentum equations in shallow waters. The connection between the generalized Boltzmann equation and classical shallow water equations provides a framework for formulating new computational approaches to surface water flows. To illustrate, a first-order explicit scheme based on the generalized Boltzmann equation for 1D shallow waters in frictionless and horizontal channels is formulated. The resulting scheme is applied to the classical dam break problem. Comparison with the analytical solution shows that the Boltzmann-based scheme is highly accurate and free of spurious oscillations, illustrating the potential of the method for surface water problems and other applications.