Numerical treatment in resonant regime for shallow water equations with discontinuous topography

This paper deals with numerical treatments for the shallow water equations with discontinuous topography when the initial data belong to both supersonic region and subsonic region. This kind of data are present in both engineering and rivers, but they are not always well-treated in existing schemes. Our goal is to improve the well-balanced scheme constructed earlier in our work by introducing a computing corrector into the construction of the scheme. First, a further study in the construction of the well-balanced scheme reveals that the errors could make the approximate states near the critical surface that ought to be in one side of the critical surface fall into the other side. This qualitative change, though small, may cause much larger errors following stationary hydraulic jumps formed from these approximate states due to the jump of the bottom. Then, we introduce a corrector in the computing algorithm that selects the equilibrium states in the construction of the well-balanced scheme such that the approximate stationary hydraulic jumps always remain in the right region. Numerical tests show that the well-balanced method using an underlying numerical flux such as Lax–Friedrichs flux, FORCE, GFORCE, or Roe fluxes can approximate very well the exact solution even when the initial data are on both supercritical region and subcritical region.