A comparison of lattice Boltzmann schemes for sub-critical shallow water flows

In this paper, we test the numerical properties of several variants of the lattice Boltzmann method (LBM) for simulating the shallow water flows. Specifically, we perform a systematic comparison of five different schemes: (i) the single-relaxation-time LBM, the (ii) raw-moments-based and (iii) central-moments-based multiple-relaxation-time LBMs, and the (iv) two-stages and (v) one-stage simplified LBMs. Concerning the latter, traditional simplified schemes require a fractional step two-stages technique. Building on the work Delgado-Gutierrez et al. ["A single-step and simplified graphics processing unit lattice Boltzmann method for high turbulent flows," Int. J. Numer. Methods Fluids 93, 2339-2361 (2021)], we derive a one-stage approach, where the procedure spans the grid points just once per time step. All the aforementioned LBMs are tested against five well-consolidated benchmark problems, and their numerical performance is assessed. Overall, populations-based schemes show superior accuracy and convergence properties. We link this behavior to the higher numerical dissipation introduced by the simplified models.