This study presents a novel, simple, but rather accurate approximation of the eigenvalues of the system formed by the Saint-Venant-Exner equations, based on the comparison between eigenvalues for the complete system and eigenvalues for the water phase only. Moreover, a strategy is proposed to compute efficiently the intercell fluxes by properly adapting a Harten, Lax, and van Leer scheme for each equation. Two examples of transient transcritical flows are developed: the erosive migration of a knickpoint induced by an increase in the bed slope, and the evolution of a hydraulic jump over a mobile bed.
Goutière, L., Soares Frazao, S., Savary, C., Laraichi, T., & Zech, Y. (2008). One-dimensional model for transient flows involving bed-load sediment transport and changes in flow regimes. Journal of Hydraulic Engineering (New York), 134(6), 726-735. https://doi.org/10.1061/(ASCE)0733-9429(2008)134:6(726) (Original work published 2008)