Modelling the transition between fixed and mobile bed conditions in two-phase free-surface flows: The Composite Riemann Problem and its numerical solution

In a two-phase free-surface flow, the transition from a mobile-bed condition to a fixed-bed one (and vice versa) occurs at a sharp interface across which the relevant system of partial differential equations changes abruptly. This leads to the possibility of conceiving a new type of Riemann Problem (RP), which we have called Composite Riemann Problem (CRP), where not only the initial constant values of the variables but also the system of equations change from left to right of a discontinuity. In this paper, we present a strategy for solving a CRP by reducing it to a standard RP of a single, composite system of equations. This can be obtained by combining the two original systems by means of a suitable weighting function, namely the erodibility variable, and the introduction of an appropriate differential equation for this quantity. In this way, the CRP problem can be analyzed theoretically with standard methods, and the features of the solutions can be clearly identified. In particular, a stationary contact wave is able to correctly describe the sharp transition between mobile- and fixed-bed conditions. A finite volume scheme based on the Multiple Averages Generalized Roe approach (Rosatti and Begnudelli (2013) [22]) was used to numerically solve the fixed-mobile CRP. Several test cases demonstrate the effectiveness, exact well balanceness and high accuracy of the scheme when applied to problems that fall within the physical range of applicability of the relevant mathematical model.