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Central-upwind schemes for the system of shallow water equations with horizontal temperature gradients

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Abstract

We introduce a central-upwind scheme for one- and two-dimensional systems of shallow-water equations with horizontal temperature gradients (the Ripa system). The scheme is well-balanced, positivity preserving and does not develop spurious pressure oscillations in the neighborhood of temperature jumps, that is, near the contact waves. Such oscillations would typically appear when a conventional Godunov-type finite volume method is applied to the Ripa system, and the nature of the oscillation is similar to the ones appearing at material interfaces in compressible multifluid computations. The idea behind the proposed approach is to utilize the interface tracking method, originally developed in Chertock et al. (M2AN Math Model Numer Anal 42:991–1019, 2008) for compressible multifluids. The resulting scheme is highly accurate, preserves two types of “lake at rest” steady states, and is oscillation free across the temperature jumps, as it is illustrated in a number of numerical experiments.

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Acknowledgments

The authors would like to warmly thank Prof. M. Lukáčová-Medvid’ová for fruitful and valuable discussions and for pointing out a fundamental mistake in a preliminary version of this paper.

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Authors and Affiliations

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

    Alina Chertock

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

    Alexander Kurganov & Yu Liu

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  1. Alina Chertock
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  2. Alexander Kurganov
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  3. Yu Liu
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Correspondence to Alina Chertock.

Additional information

The work of A. Chertock was supported in part by the NSF Grants DMS-1115682 and DMS-1216974 and by the ONR Grant N00014-12-1-0832. The research of A. Kurganov was supported in part by the NSF Grants DMS-1115718 and DMS-1216957 and by the ONR Grant N00014-12-1-0833.

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Chertock, A., Kurganov, A. & Liu, Y. Central-upwind schemes for the system of shallow water equations with horizontal temperature gradients. Numer. Math. 127, 595–639 (2014). https://doi.org/10.1007/s00211-013-0597-6

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  • DOI: https://doi.org/10.1007/s00211-013-0597-6

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