Simple waves and the Riemann problem in compressible shallow water flows

Compressible gas flows in the gravity field above a homogeneous surface in shallow water approximation are investigated. All the self-similar continuous and discontinuous solutions are derived. A solution of the basic initial discontinuity decay problem for the compressible shallow water equations above a homogeneous surface is provided. It is shown that the solution consists of four different configurations that depend on the initial conditions. The necessary and sufficient conditions for the realization of each configuration are derived. (paper)