A lattice Boltzmann model for the viscous shallow water equations with source terms

•A lattice Boltzmann model is proposed for a class of viscous shallow water equations.•The effective rainfall has been considered in the viscous shallow water equations.•The governing equations can be correctly recovered from the proposed model.•The second-order moment of source terms can effectively ensure the conservation. In this paper, a lattice Boltzmann (LB) model is proposed for a class of viscous shallow water equations, in which a second-order moment of the source term is applied to recover the viscosity in the governing equation and eliminate the additional errors generated during the Chapman-Enskog analysis. There are three different schemes based on different treatments of the source term. Through numerical simulations of several classical benchmark problems, we find that the second-order moment of the source term can not only improve the accuracy of the LB model, but also ensure the conservation of the system. In addition, the influence of rainfall intensity on shallow water flow is also taken into account, and the results show that present LB model can also accurately study such problems as overland flows.