  Geometric discretizations that preserve certain Hamiltonian structures at the
discrete level has been proven to enhance the accuracy of numerical schemes. In
particular, numerous symplectic and multi-symplectic schemes have been proposed
to solve numerically the celebrated Korteweg-de Vries (KdV) equation. In this
work, we show that geometrical schemes are as much robust and accurate as
Fourier-type pseudo-spectral methods for computing the long-time KdV dynamics,
and thus more suitable to model complex nonlinear wave phenomena.
