The LHLLC scheme for Two-Layer and Two-Phase transcritical flows over a mobile bed with avalanching, wetting and drying

•A minimally diffusive scheme, LHLLC, is presented for debris flow equations.•The model is applicable to hyper-concentrated sediment loads.•Stationarity is preserved in problems with a sloping bed and wet/dry interfaces.•The scheme can be applied to Two-Phase and Two-Layer debris flow equations.•An avalanching scheme is used to account for gravitational slumping.

A numerical method of the Godunov type is presented for solving either Two-Phase or Two-Layer forms of Debris Flow Models (DFMs) describing shallow-water flow and sediment dynamics. DFMs explicitly link sediment concentrations to the momentum balance, and thus can be applied to cases involving high sediment concentrations, as in debris flows, in addition to low concentration test cases typically found in surface waters. In this paper, Two-Phase and Two-Layer DFMs are presented in a common mathematical framework to illuminate key similarities and differences and lay a foundation for a general purpose DFM solver. The proposed solver termed LHLLC is shown to achieve good accuracy over a wide range of test cases. Importantly, numerical diffusion of sediment profiles is minimized, particularly on steep slopes, the scheme is shown to preserve stationary solutions involving wet/dry interfaces, and the scheme accounts for gravity-driven slumping (avalanching) which cannot be resolved by classical DFMs.