Paper detail

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

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $ε^2$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length 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 $ε$ within the `interior' of the Whitham oscillatory zone, of order $ε^{1/3}$ at the left boundary outside the Whitham zone and of order $ε^{1/2}$ at the right boundary outside the Whitham zone.