The role of the Rankine-Hugoniot relations in staggered finite difference schemes for the shallow water equations

The purpose of this work is to point out the relevance of the Rankine-Hugoniot jump relations regarding the numerical solution of the inviscid shallow water equations. To arrive at physically relevant solutions in rapidly varied flow, it is of crucial importance that continuity of mass flux and momentum flux across a steady discontinuity is fulfilled at the discrete level. By adopting this viewpoint, finite difference schemes can be studied that may be well suited to solve shallow water flow problems involving discontinuities, while they are not based on a characteristic decomposition of the governed hyperbolic equations. Three schemes on staggered grids with either the water level or the water depth at the cell centre and the flow velocity or the depth-integrated velocity at the cell interface are examined. They differ in (1) the character of the transport velocity to bias the discretization of the advective acceleration term in the upwind direction, and (2) the determination of the water depth at the cell face with which the depth-integrated velocity must be linked to the flow velocity. A detailed analysis is provided and aimed at highlighting the necessity of fulfilling the Rankine-Hugoniot jump conditions for preventing the odd-even decoupling problem. The accuracy and robustness of three selected schemes is assessed by means of convergence tests, three idealized 1D test problems with exact solutions and a 1D laboratory experiment of the breaking, runup and rundown of a solitary wave on a sloping beach. Numerical results reveal that schemes satisfying exactly the jump conditions display improved performance over schemes which do not share this property. Also, these results support strong evidence on the link between not fulfilling the jump conditions and the appearance of odd-even oscillations.