  In this paper, we consider the evolution of spacelike graphic curves defined
over a piece of hyperbola $\mathscr{H}^{1}(1)$, of center at origin and radius
$1$, in the $2$ dimensional Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$ along
an anisotropic 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 curves
converge smoothly to a piece of hyperbola of center at origin and prescribed
radius, which actually corresponds to a constant function defined over the
piece of $\mathscr{H}^{1}(1)$, as time tends to infinity.
