ИСТИНА

The shallow water flows over an arbitrary bottom with discontinuities is considered. The complex topography is approximated by a piecewise continuous linear function on unstructured triangular-quadrilateral meshes, which gives the opportunity to resolve sharp changes in the topography. The solution of the shallow water equations is based on a Godunov-type scheme, which includes the stage of the Riemann problem solution. In some cases, the exact solution of the Riemann problem over bottom discontinuity is not unique, which complicates the process of applying the exact solution in Godunov type schemes. However, it is shown that this issue can be overcome by involving an additional hypothesis which ensures the property of well-posedness of the problem. In the new formulation examples of all possible configurations of the exact solution are given. Other ways to overcome this problem are discussed. The numerical simulations of one- and two-dimensional model tests are presented, as well as calculations of the real flows. The work was supported by the Russian Science Foundation, project no. 17-77-30006.