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BIT Numerical Mathematics
Erschienen in: BIT Numerical Mathematics 1/2017

20.07.2016

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

verfasst von: G. M. Coclite, J. Ridder, N. H. Risebro

Erschienen in: BIT Numerical Mathematics | Ausgabe 1/2017

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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.
Vorheriger Artikel On estimating the separation of two periodic matrix sequences
Nächster Artikel Locally Linearized methods for the simulation of stochastic oscillators driven by random forces

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Fußnoten
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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Metadaten
Titel
A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain
verfasst von
G. M. Coclite
J. Ridder
N. H. Risebro
Publikationsdatum
20.07.2016
Verlag
Springer Netherlands
Erschienen in
BIT Numerical Mathematics / Ausgabe 1/2017
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-016-0625-x

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