Exact Riemann solutions to shallow water equations

Abstract

We determine completely the Riemann solutions to the shallow water equations with a bottom step, including the dry bed problem. The nonstrict hyperbolicity of this first-order system of partial differential equations leads to resonant waves and nonunique solutions. To address these difficulties we construct the L–M and R–M curves in the state space. For the bottom step elevated from left to right, we classify the L–M curve into five different cases and the R–M curve into two different cases based on the subcritical and supercritical Froude number of the Riemann initial data as well as the jump of the bottom step. The behaviors of all basic cases of the L–M and R–M curves are fully analyzed. We observe that the non–uniqueness of the Riemann solutions is due to bifurcations on the L–M or R–M curves. The possible solutions including classical waves and resonant waves as well as dry bed state are solved in a uniform framework for any given Riemann initial data.