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Unconditionally Stable Splitting Methods for the Shallow Water Equations

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Abstract

The front-tracking method for hyperbolic conservation laws is combined with operator splitting to study the shallow water equations. Furthermore, the method includes adaptive grid refinement in multidimensions and shock tracking in one dimension. The front-tracking method is unconditionally stable, but for practical computations feasible CFL numbers are moderately above unity (typically between 1 and 5). The method resolves shocks sharply and is highly efficient. The numerical technique is applied to four test cases, the first being an expanding bore with rotational symmetry. The second problem addresses the question of describing the time development of two constant water levels separated by a dam that breaks instantaneously. The third problem compares the front-tracking method with an explicit analytic solution of water waves rotating over a parabolic bottom profile. Finally, we study flow over an obstacle in one dimension.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway. email: holdahl@math.ntnu.no

    Runar Holdahl & Helge Holden

  2. Department of Informatics, University of Oslo, P.O. Box 1080 Blindern, NO-0316, Oslo, Norway. email: kalie@ifi.uio.no

    Knut-Andreas Lie

Authors
  1. Runar Holdahl
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  2. Helge Holden
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  3. Knut-Andreas Lie
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Holdahl, R., Holden, H. & Lie, KA. Unconditionally Stable Splitting Methods for the Shallow Water Equations. BIT Numerical Mathematics 39, 451–472 (1999). https://doi.org/10.1023/A:1022366502335

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