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A new second-order modified hydrostatic reconstruction for the shallow water flows with a discontinuous topography. (English) Zbl 07310825
The authors present a new effective and reliable modified second-order HR (hydrostatic reconstruction) scheme for the shallow water equations with discontinuous bottom topography based on the water surface level, the depth-averaged velocity, and the bottom topography. The reconstruction at the wet-dry front is modified to maintain the stationary solution and to preserve the non-negative water depth. In order to correctly reflect the acceleration due to the sloped bottom, the intermediate bottom level is redefined at the cell interface. The discretization of the source term is also modified when the fluid has enough mechanical energy to climb the step. The new scheme is proved to be exactly well-balanced and positivity-preserving. In particular, this scheme can correctly reflect the wave pattern for the shallow downhill flow over steps.
MSC:
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B09 Positive solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
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