Re-brightening bumps are frequently observed in gamma-ray burst 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. This approach is a semi-analytic method and is very convenient. Our results show that late high-amplitude bumps cannot be produced under common conditions, rather only a short plateau may emerge even when the encounter occurs at an early time (104?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 caused by other scenarios.
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