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

Source abstract

Abstract The Cauchy problem for the Korteweg—de Vries (KdV) equation with small dispersion of order ϵ 2 , ϵ ≪ 1, is characterized by the appearance of a zone of rapid, modulated oscillations of wavelength of order ϵ. 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 ϵ between 10 −1 and 10 −3 . The numerical results are compatible with a difference of order ϵ close to the center within the “interior” of the Whitham oscillatory zone, of order ϵ 1/3 at the left boundary outside the Whitham zone and of order √ϵ at the right boundary outside the Whitham zone. © 2007 Wiley Periodicals, Inc.