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Central-upwind scheme for shallow water equations with discontinuous bottom topography

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Andrew Bernstein & Alina Chertock

  2. Mathematics Department, Tulane University, New Orleans, LA, 70118, USA

    Alexander Kurganov

Authors
  1. Andrew Bernstein
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  2. Alina Chertock
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  3. Alexander Kurganov
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Correspondence to Alexander Kurganov.

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Bernstein, A., Chertock, A. & Kurganov, A. Central-upwind scheme for shallow water equations with discontinuous bottom topography. Bull Braz Math Soc, New Series 47, 91–103 (2016). https://doi.org/10.1007/s00574-016-0124-3

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  • DOI: https://doi.org/10.1007/s00574-016-0124-3

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