  We discuss a new numerical schema for solving the initial value problem for
the Korteweg-de Vries equation for large times. Our approach is based upon the
Inverse Scattering Transform that reduces the problem to calculating the
reflection coefficient of the corresponding Schr\"odinger equation. Using a
step-like approximation of the initial profile and a fragmentation principle
for the scattering data, we obtain an explicit recursion formula for computing
the reflection coefficient, yielding a high resolution KdV solver. We also
discuss some generalizations of this algorithm and how it might be improved by
using Haar and other wavelets.
