A note on the Riemann problem for shallow water equations with discontinuous topography

Abstract

In a recent paper LeFloch and Thanh (2011) constructed the Riemann solutions for shallow water equations with discontinuous topography, where the stationary wave is determined by solving steady state equations. Here we consider the case when the original steady equations do not have a solution automatically. We thus construct the supplement Riemann solutions and prove that a backward shock wave emits to decelerate the flow which makes the steady equations solvable. Global entropy condition is imposed subsequently to select the physical relevant solution which states that the defined energy function should not only increase but increase to the maximum rate. Numerical results are given to verify our analysis.