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

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• 作者：G. M. Coclite ; J. Ridder ; N. H. Risebro
• 关键词：Ostrovsky ; Hunter equation ; Short ; pulse equation ; Vakhnenko equation ; Finite difference methods ; Monotone scheme ; Existence ; Uniqueness ; Stability ; Convergence ; Entropy solution
• 刊名：BIT Numerical Mathematics
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：57
• 期：1
• 页码：93-122
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Computational Mathematics and Numerical Analysis; Numeric Computing; Mathematics, general;
• 出版者：Springer Netherlands
• ISSN：1572-9125
• 卷排序：57

文摘

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.