Exact solutions to the Riemann problem of the shallow water equations with a bottom step

The similarity solution to the Riemann problem of the one dimensional shallow water equations (SWE) with a bottom step discontinuity is presented. The step is placed at the same location where the how variables are initially discontinuous. While the solutions found are still a superposition of travelling waves belonging to the two well-known families of the shallow water system, namely hydraulic jumps and rarefactions, the appearance of a standing discontinuity at the step position produces a very interesting solution pattern. This is mainly due to the asymmetry introduced by the step. The adopted solution procedure combines the basic theory of hyperbolic systems of conservation laws together with a sound interpretation of the physical concepts embedded in the shallow water system. This finally leads to a set of algebraic equations that must be iteratively solved. The ideas contained in this paper may be of valuable help to the understanding of the behaviour of the SWE with source terms, that constitute the core of many mathematical models for free surface flow simulation. (C) 2001 Elsevier Science Ltd. All rights reserved.