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 equation. In this work, we show that geometrical schemes are as much robust and accurate as Fourier-type pseudospectral methods for computing the long-time KdV dynamics, and thus more suitable to model complex nonlinear wave phenomena.