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 ǫ, ǫ ≪ 1, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order ǫ. 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 ǫ between 10 and 10. The numerical results are compatible with a difference of order ǫ within the ‘interior’ of the Whitham oscillatory zone, of order ǫ 1 3 at the left boundary outside the Whitham zone and of order √ ǫ at the right boundary outside the Whitham zone.