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Central-upwind scheme for shallow water equations with discontinuous bottom topography. (English) Zbl 1338.76072

Summary: Finite-volume central-upwind schemes for shallow water equations were proposed in [the last author and G. Petrova, Commun. Math. Sci. 5, No. 1, 133–160 (2007; Zbl 1226.76008)]. These schemes are capable of maintaining “lake-at-rest” steady states and preserving the positivity of the computed water depth. The well-balanced and positivity preserving features of the central-upwind schemes are achieved, in particular, by using continuous piecewise linear interpolation of the bottom topography function. However, when the bottom function is discontinuous or a model with a moving bottom topography is studied, the continuous piecewise linear approximationmay not be sufficiently accurate and robust.
In this paper, we modify the central-upwind scheme by approximating the bottom topography function using a discontinuous piecewise linear reconstruction (the same approximation used to reconstruct evolved quantities in the finite-volume setting) as well as implementing a special quadrature for the geometric source term and draining time step technique. We prove that the new central-upwind scheme possesses the wellbalanced and positivitypreserving properties and illustrate its performance on a number of numerical examples.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
86-08 Computational methods for problems pertaining to geophysics
86A05 Hydrology, hydrography, oceanography
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics

Citations:

Zbl 1226.76008
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References:

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