A well-balanced lattice Boltzmann model for the depth-averaged advection–diffusion equation with variable water depth

Abstract A well-balanced lattice Boltzmann model for the depth-averaged advection–diffusion equation is developed to solve the solute transport problem in shallow water flows with obvious gradient changes in water depth. The flow field is simulated by the lattice Boltzmann model for shallow water equations with the central moments (CMs) collision operator and D2Q9 lattice pattern. In this model, the diffusion term of the depth-averaged advection–diffusion equation is divided into two parts by the product rule, one part of which is combined into the advection term to produce the pseudo velocity. The relaxation time is independent of the diffusion coefficient, and the remaining value of ( 3 + 3 ) ∕ 6 derived from the Chapmann–Enskog expansion reduces the third-order error. According to the different treatments of the water depth gradient in the equilibrium distribution function, center scheme and linked scheme are developed. Both schemes can ensure the conservation of the solute mass. Numerical tests show that these two schemes give predictions in good agreement, but the linked scheme works better at the breakpoint. Comparisons with the reference indicate that this model is advantageous with respect to both stability and efficiency.