Generalized transformation of the lattice Boltzmann method for shallow water flows

Abstract

A one-dimensional lattice Boltzmann model is developed to solve the shallow water equations for steady and unsteady flows within both the subcritical and supercritical regimes. Previous work is extended through a generalized Galilean transformation applied to the standard scheme. The transformation yields a general asymmetric lattice Boltzmann model scheme which can successfully model a wide range of both subcritical and supercritical flow regimes, and enables implementation of the asymmetric model for practical purposes. In current work, a new set of equilibrium functions, boundary conditions and the external force weights are derived for the generalized transformed scheme. A new stability region is also defined, allowing selection of a lattice speed that maintains numerical stability for a wider range of sub-and supercritical flows and combinations of those flow conditions, compared to the previous scheme with fixed asymmetry. The model is validated against a range of benchmark cases in open-channel hydraulics that demonstrate the applicability of the new model.

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