A Godunov method is proposed for the computation of open-channel flows in conditions of rapid bed erosion and intense sediment transport. Generalized shallow water equations govern the evolution of three distinct interfaces: the water free-surface, the boundary between pure water and a sediment transport layer, and the morphodynamic bottom profile. Based on the HLL scheme of Harten, Lax and Van Leer (1983), a finite volume numerical solver is constructed, then extended to second-order accuracy using Strang splitting and MUSCL extrapolation. Lateralisation of the momentum flux is adopted to handle the non-conservative product associated with bottom slope. Computational results for erosional dam-break waves are compared with experimental measurements and semi-analytical Riemann solutions. Copyright © 2003 John Wiley & Sons, Ltd.