Finite-volume central-upwind schemes for shallow water equations were proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), 133–160]. 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.
hyperbolic system of conservation and balance laws semi-discrete centralupwind scheme Saint Venant system of shallow water equations
Mathematical subject classification
76M12 65M08 35L65 86-08 86A05
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