Preprint 99/2005

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

Tamara Grava and Christian Klein

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Submission date: 14. Nov. 2005
Pages: 41
published in: Communications on pure and applied mathematics, 60 (2007) 11, p. 1623-1664 
DOI number (of the published article): 10.1002/cpa.20183
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Abstract:
The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order formula15, formula17, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order formula19. 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 formula19 between formula23 and formula25. The numerical results are compatible with a difference of order formula19 within the `interior' of the Whitham oscillatory zone, of order formula29 at the left boundary outside the Whitham zone and of order formula31 at the right boundary outside the Whitham zone.

19.04.2013, 01:41