Introduction

The exact solution of the one-dimensional problem in shallow water approximation with initial conditions shown in the following figure is sought (arbitrary bottom discontinuity, depth and velocity to the left and to the right of it are given).

Interest in the Riemann problem is associated with the use of its solver in numerical simulation of flows based on the Godunov method, which in various modifications is widely and successfully used to solve hyperbolic systems of equations describing flows in the shallow water approximation, in gas dynamics, in magnetic hydrodynamics. In general, it is known, that the solution of the Riemann problem for shallow water equations with a bottom discontinuity is not unique; that makes it difficult to apply an exact solver of this problem in numerical methods, since it is unclear which solution should be chosen. The solution to this problem is proposed in the work [A. I. Aleksyuk, V. V. Belikov, J. Comput. Phys., 2019]. It is shown that all the "extra" solutions do not satisfy the requirement of continuous dependence of the discharge at the bottom discontinuity on the initial conditions. Thus, the condition that the discharge is continuous at a jump from the initial conditions makes the problem statement correct – for the first time, it is proved that there is a unique solution to such a problem.

In the proof it is shown how to switch correctly between possible solutions at bifurcation points when flow regimes change. An algorithm (solver) is developed for obtaining an exact solution to the Riemann problem with bottom discontinuity for arbitrary initial conditions. All possible configurations of the solution were found, and the new solver was tested. The practical value of the work is related to the possibility of using a new solver in numerical methods (of the Godunov type) for calculating flows across the boundaries of cells in the computational grid. The new solver is more effective than traditional approaches to modeling flows with a complex bottom topography (including the one based on the Riemann problem solver over a continuous bottom). Thus, it is shown that the results of a numerical simulation of stationary flows with regions of sharply changing bottom marks resolved on coarse grids correspond to exact solutions and can be reproduced using traditional numerical algorithms only on detailed grids. The developed solver was integrated into the author's software package STREAM 2D for simulating applied problems in the approximation of shallow water with complex topography.

Related publications: [A.I. Aleksyuk, V.V. Belikov, JCP, 2019; A.I. Aleksyuk et al., JCP, 2022]

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