01358nas a2200109 4500008004300000245009600043210006900139520096500208100001801173700002101191856003601212 2007 en_Ud 00aNumerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations0 aNumerical solution of the small dispersion limit of Korteweg de 3 aThe Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\\\\epsilon^2$, 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 $\\\\epsilon^{1/3}$ at the left boundary outside the Whitham zone and of order $\\\\epsilon^{1/2}$ at the right boundary outside the Whitham zone.1 aGrava, Tamara1 aKlein, Christian uhttp://hdl.handle.net/1963/1788