Numerical solution of the discontinuous-bottom Shallow-water Equations with hydrostatic pressure distribution at the step

Free-surface flows are usually modelled by means of the Shallow-water Equations: this system of hyperbolic equations 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 fluid mass. Recently, much attention has been paid to the case in which bottom discontinuities are present in the physical domain to be represented: in this case, it is difficult to define the non-conservative product in the distributional sense. Here, the discontinuous-bottom Shallow-water Equations with hydrostatic pressure distribution at the bed step (Bernetti et al., 2006) are discussed in the context of the theory of Dal Maso et al. (1995) [9]; finally, a first-order numerical scheme is presented, which is consistent for regular solutions, and which is able to capture contact discontinuities at bottom steps. Numerous tests are presented to show the feasibility of the scheme and its ability to converge to the exact solution in the cases of smooth as well as discontinuous bed profiles.

► The discontinuous bottom SWE are considered, with hydrostatic pressure at the step. ► Weak solutions are defined considering the DLM theory. ► The chosen family of paths is significant of the pressure distribution on the step. ► The Hugoniot curve across the bed step is presented in an operative form. ► The numerical solution is obtained using a hydrostatic reconstruction technique.