  We study numerically the small dispersion limit for the Korteweg-de Vries
(KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ for $\epsilon\ll1$ and give a
quantitative comparison of the numerical solution with various asymptotic
formulae for small $\epsilon$ in the whole $(x,t)$-plane. The matching of the
asymptotic solutions is studied numerically.
