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A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain

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Abstract

We prove the 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 periodic boundary conditions. The equation models, for example, shallow water waves in a rotating fluid and ultra-short light pulses in optical fibres, and its solutions can develop discontinuities in finite time. Our scheme extends monotone schemes for conservation laws to this equation. The convergence result also provides an existence proof for periodic entropy solutions of the general Ostrovsky-Hunter equation. Uniqueness and an \({\mathscr {O}}({\varDelta x}^{1/2})\) bound on the \(L^1\) error of the numerical solutions are shown using Kružkov’s technique of doubling of variables and a “Kuznetsov type” lemma. Numerical examples confirm the convergence results.

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Notes

  1. In principle, a suitable CFL number can be determined from (10). Here we have \(||f'(u_0) ||_{L^\infty (0,1)}=1/36\) for the first experiment and \(||f'(u_0) ||_{L^\infty (0,1)}=1/20\) for the second, therefore \(\lambda =\varDelta t/\varDelta x\) should satisfy \(\lambda \le 36\) and \(\lambda \le 20\), respectively. However, since the \(L^\infty \) bound from Lemma 1 allows \(||u^n ||_{\infty }\) to grow, it can be necessary to choose a smaller \(\lambda \).

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Authors and Affiliations

  1. Department of Mathematics, University of Bari, Via E. Orabona 4, 70125, Bari, Italy

    G. M. Coclite

  2. Department of Mathematics, University of Oslo, P.O.Box 1053, Blindern, 0316, Oslo, Norway

    J. Ridder & N. H. Risebro

Authors
  1. G. M. Coclite
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  2. J. Ridder
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  3. N. H. Risebro
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Correspondence to J. Ridder.

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Communicated by Jan Hesthaven.

G. M. Coclite is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The work of J. Ridder and N. H. Risebro is supported by the Research Council of Norway, project number 214495.

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Coclite, G.M., Ridder, J. & Risebro, N.H. A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain. Bit Numer Math 57, 93–122 (2017). https://doi.org/10.1007/s10543-016-0625-x

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