Discrete Boltzmann model of shallow water equations with polynomial   equilibria
  A type of discrete Boltzmann model for simulating shallow water flows is derived by using the Hermite expansion approach. Through analytical analysis, we study the impact of truncating distribution function and discretizing particle velocity space. It is found that the convergence behavior of expansion is nontrivial while the conservation laws are naturally satisfied. Moreover, the balance of source terms and flux terms for steady solutions is not sacrificed. Further numerical validations show that the capability of simulating supercritical flows is enhanced by employing higher order expansion and quadrature. 
Jianping Meng and Xiao-Jun Gu and David R Emerson and Yong Peng and   Jianmin Zhang
physics.flu-dyn