  In this paper, we consider the evolution of spacelike graphic hypersurfaces
defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center
at origin and radius $1$, 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 hyperbolic plane of
center at origin and prescribed radius, which actually corresponds to a
constant function defined over the piece of $\mathscr{H}^{n}(1)$, as time tends
to infinity.
