Two-relaxation-times Lattice Boltzmann schemes for solute transport in unsaturated water flow, with a focus on stability

Abstract

We develop two-relaxation-times Lattice Boltzmann schemes (TRT) with two relaxation functions Λ ± ( r → , t ) for solving highly non-linear equations for groundwater modeling in d-dimensions, namely, the Richards equation for water content distribution θ ( r → , t ) in unsaturated flow and the associated transport equation for solute concentration C ( r → , t ) , advected by the local Darcian water flux. The method is verified against the analytical solutions and the HYDRUS code where the TRT schemes behave more robustly for small diffusion coefficients and sharp infiltration profiles. The focus is on the stability and efficiency of two transport schemes. The first scheme conventionally prescribes C for diffusive flux equilibrium variable while conserving θC. The second scheme prescribes θC for both variables, expecting to retain the stable parameter areas and velocity amplitudes recently predicted by linear von Neumann stability analysis. We show that the first scheme reduces the stable diffusion range, e.g. from Λ −/ d to θΛ −/ d for simplest velocity sets, but it also modifies the linearized numerical diffusion, from − Λ − U α U β to − θΛ − U α U β , giving rise to possible enhancement of stable velocity U 2, max by a factor 1/ θ. This analysis indicates that the first scheme is most efficient for infiltration into dry soil. When the product Λ + Λ − is kept constant, we find a good agreement between the attainable velocity and our predictions providing that Λ − does not exceed ≈5. Otherwise, approaching two opposite stability limits, Λ + → 0 when Λ − → ∞ , the stable velocity amplitude drastically falls for the two transport TRT schemes. At the same time, their BGK submodels Λ + = Λ − may keep the optimal stability for diffusion-dominant problems but their boundary and bulk approximations are completely destroyed. The analysis presented here may serve as a starting point for construction of the suitable equilibrium transformations, based on the analytical stability argument and a proper parameter choice.