Generalized Roe schemes for 1D two-phase, free-surface flows over a mobile bed

The problem of two-phase, free-surface flows over a mobile bed is characterized by a hyperbolic partial differential equations system that shows nonconservative terms and highly nonlinear relations between primitive and conserved variables. Weak solutions of the present problem were obtained resorting both to the distribution theory and to the integral formulation of momentum conservation: the comparison of these two approaches allowed us to give a physical insight into the meaning of the nonconservative term across a discontinuity. Starting from this result, we derived the conditions necessary to obtain generalized, well-balanced Roe solvers without using the concept of a family of paths. Two numerical schemes based on the same set of matrices have been developed, one in terms of conserved variables and one in terms of primitive variables. The friction-source term has also been included by using an upwind approach. The capabilities and limits of the proposed schemes have been analyzed by comparison with exact solutions of Riemann problems and with numerical solutions obtained with the AWB-3SRS scheme.