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Exact solutions to the Riemann problem of the shallow water equations with a bottom step. (English) Zbl 1048.76008
Summary: We present a similarity solution to Riemann problem for one-dimensional shallow water equations with a bottom step discontinuity. The step is placed at the same location where the flow variables are initially discontinuous. While the solutions found are still a superposition of travelling waves belonging to 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 physical concepts embedded in the shallow water system. This finally leads to a set of algebraic equations that must be iteratively solved.

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M55 Dimensional analysis and similarity applied to problems in fluid mechanics
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