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

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 op...

Full description

Saved in:
Bibliographic Details
Published inBIT Vol. 57; no. 1; pp. 93 - 122
Main Authors Coclite, G. M., Ridder, J., Risebro, N. H.
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.03.2017
Springer Nature B.V
Subjects
Boundary conditions
Computational Mathematics and Numerical Analysis
Mathematics
Mathematics and Statistics
Numeric Computing
Stability
65M06
Uniqueness
35L35
Monotone scheme
35M33
Vakhnenko equation
Short-pulse equation
Convergence
Ostrovsky-Hunter equation
Existence
Finite difference methods
Entropy solution
Online AccessGet full text

Cover

More Information
Summary: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 O ( Δ 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.
ISSN:0006-3835
1572-9125
DOI:10.1007/s10543-016-0625-x