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A convergent finite difference scheme for the Ostrovsky–Hunter equation with Dirichlet boundary conditions

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Abstract

We prove convergence of a finite difference scheme to the unique entropy solution of a general form of the Ostrovsky–Hunter equation on a bounded domain with non-homogeneous Dirichlet boundary conditions. Our scheme is an extension of monotone schemes for conservation laws to the equation at hand. The convergence result at the center of this article also proves existence of entropy solutions for the initial-boundary value problem for the general Ostrovsky–Hunter equation. Additionally, we show uniqueness using Kružkov’s doubling of variables technique. We also include numerical examples to confirm the convergence results and determine rates of convergence experimentally.

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Notes

  1. Here, we have and therefore \(\lambda =\Delta t/\Delta x\) should satisfy \(\lambda \le 36\). However, since the bound from Lemma 1 allows for some growth of choosing a smaller \(\lambda \) can be neccessary.

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Acknowledgements

The authors want to thank Nils Henrik Risebro from the University of Oslo for many insightful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, University Park, USA

    J. Ridder

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    A. M. Ruf

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  1. J. Ridder
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Correspondence to A. M. Ruf.

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Communicated by Jan Nordström.

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The second author has received funding from the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska-Curie Grant Agreement No. 642768.

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Ridder, J., Ruf, A.M. A convergent finite difference scheme for the Ostrovsky–Hunter equation with Dirichlet boundary conditions. Bit Numer Math 59, 775–796 (2019). https://doi.org/10.1007/s10543-019-00746-7

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