The Analytical and Numerical Solutions of the Shallow Water Equations with the Concentrated Discontinuities in the Bed

In this article the influence of the concentrated bed discontinuities (i.e. raising or falling steps) on the
solution of the Reimann problem in the shallow water equations is studied. First, the analytical
solution of one-dimensional dambreak problem, using the mass-momentum equations over a
continuous flat and frictionless bed is reviewed. Next this solution is extended to a discontinuous bed
(i. e. a step), using the mass momentum equations at the continuous places and the mass-energy
equations at the discontinuities. Then, for the first time, the dambreak problem over a rectangular
bump is solved, analytically. In the next section, the mass-momentum equations are numerically
solved over the bump. For the discretisation of these equations over the flat continuous bed, the finite
volume Roe-TVD scheme with a minmod slope limiter is applied. At the discontinuities, however, the
mass-energy equations are discretised. The numerical results are found close to the analytical solution.
Finally, the dam break problem in a cavity is similarly investigated both analytically and numerically.