Approximate solution of discontinous-bottom shallow-water equations with assigned pressure distribution

Free-surface shallow flows can be effectively modelled by means of the Shallow-water Equations, which are usually written considering the hypothesis of gradually varied flow, hydrostatic pressure distribution and small bed slope: the system of hyperbolic equations obtained exhibits a source term which is proportional to the product of the water depth by the bed slope, and which takes into account the effect of gravity onto the fluid mass. Recently, much attention has been paid to the case that bottom discontinuities are present into the physical domain to be represented: in this case, the non-conservative product is hard to be defined in the distributional sense. Here, following the theory by Dal Maso, LeFloch and Murat [7], a numerical scheme is presented, which is well balanced and able to capture contact discontinuities due to bottom steps where the thrust exerted over the fluid is supposed to be known ([3], [18]). Numerous tests are presented, in order to show the feasibility of the scheme, and its ability to converge to the exact solution in both the cases of smooth or discontinuous bed profile