The Riemann Problem for the Dispersive Nonlinear Shallow Water Equations
by
Giovanna Grosso
DICAT, University of Genova, via Montallegro 1, 16145 Genova
Coauthors: Matteo Antuono Maurizio Brocchini

The complete analytical solution of the Riemann problem is given for the homogeneous Dispersive Nonlinear Shallow Water Equations of Antuono, Liapidevskii and Brocchini (2008), a set of hyperbolic dispersive depth- averaged equations modelling the hydrodynamics of the nearshore zone. The Riemann problem is solved for both wet- and dry-bed conditions and all the possible wave configurations of the solution are analyzed in detail. An interesting resonance phenomenon, which can occur in the wave pattern of the solution (compound wave), when one of the eigenvalues of the system becomes zero at a certain state, is also investigated. Starting from the work of Isaacson and Temple (Journal of Applied Mathematics, 1992), we define the conditions for the occurrence of the resonance and the algorithm to catch and represent it. Finally, the proposed analytical solution is compared with accurate numerical solutions and its validity is confirmed by the good agreement with the latters. The presented exact solution of the Riemann problem is essential for the construction of a Godunov type solver for the Dispersive Nonlinear Shallow Water.

Date received: March 26, 2008