  In this paper, we consider the evolution of spacelike graphic hypersurfaces
defined over a piece of de Sitter space $\mathbb{S}^{n}_{1}$ in the
$(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along the
inverse mean curvature flow with the vanishing Neumann boundary condition, and
prove that this flow exists for all the time. Moreover, we can show that, after
suitable rescaling, the evolving spacelike graphic hypersurfaces converge
smoothly to a piece of pseudosphere of prescribed radius, which corresponds to
a constant function defined over the piece of $\mathbb{S}^{n}_{1}$, as time
tends to infinity.
