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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
99/2005

Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations

Tamara Grava and Christian Klein

Abstract

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\epsilon^2$, $\epsilon\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $\epsilon$.

These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. In this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the KdV equation in the small dispersion limit and the corresponding approximate solution for values of $\epsilon$ between $10^{-1}$ and $10^{-3}$.

The numerical results are compatible with a difference of order $\epsilon$ within the "interior" of the Whitham oscillatory zone, of order $\epsilon^{\frac{1}{3}}$ at the left boundary outside the Whitham zone and of order $\sqrt{\epsilon}$ at the right boundary outside the Whitham zone.

Received:
Nov 14, 2005
Published:
Nov 14, 2005

Related publications

inJournal
2007 Repository Open Access
Tamara Grava and Christian Klein

Numerical solution of the small dispersion limit of Korteweg-deVries and Whitham equations

In: Communications on pure and applied mathematics, 60 (2007) 11, pp. 1623-1664