We provide some numerical computations for the soliton dynamics of the
nonlinear Schr\"odinger equation with an external potential. After computing
the ground state solution $r$ of a related elliptic equation we show that, in
the semi-classical regime, the center of mass of the solution with initial
datum modelled on $r$ is driven by the solution of a Newtonian type law.
Finally, we provide some examples and analyze the numerical errors in the two
dimensional case when $V$ is an harmonic potential.