  Re-brightening bumps are frequently observed in gamma-ray burst (GRB)
afterglows. Many scenarios have been proposed to interpret the origin of these
bumps, of which a blast wave encountering a density-jump in the circumburst
environment has been questioned by recent works. We develop a set of
differential equations to calculate the relativistic outflow encountering the
density-jump by extending the work of Huang et al. (1999). This approach is a
semi-analytic method and is very convenient. Our results show that late
high-amplitude bumps can not be produced under common conditions, only short
plateau may emerge even when the encounter occurs at early time ($< 10^4$ s).
In general, our results disfavor the density-jump origin for those observed
bumps, which is consistent with the conclusion drawn from full hydrodynamics
studies. The bumps thus should be due to other scenarios.
