TY - RPRT
T1 - Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations
Y1 - 2007
A1 - Tamara Grava
A1 - Christian Klein
AB - The 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.
JF - Comm. Pure Appl. Math. 60 (2007) 1623-1664
UR - http://hdl.handle.net/1963/1788
U1 - 2756
U2 - Mathematics
U3 - Mathematical Physics
ER -