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Direct multi-dimensional numerical simulation is the most reliable approachfor calculating the fluid dynamics and observational signatures of relativisticjets in gamma-ray bursts (GRBs). We present a two-dimensional relativistichydrodynamic simulation of a GRB outflow during the afterglow phase, which usesthe fifth-order weighted essentially non-oscillatory scheme and adaptive meshrefinement. Initially, the jet has a Lorentz factor of 20. We have followed itsevolution up to 150 years. Using the hydrodynamic data, we calculatesynchrotron radiation based upon standard afterglow models and compare ourresults with previous analytic work. We find that the sideways expansion of arelativistic GRB jet is a very slow process and previous analytic works haveoverestimated its rate. In our computed lightcurves, a very sharp jet break isseen and the post-break lightcurves are steeper than analytic predictions. Wefind that the jet break in GRB afterglow lightcurves is mainly caused by themissing flux when the edge of the jet is observed. The outflow becomesnonrelativistic at the end of the Blandford-McKee phase. But it is still highlynonspherical, and it takes a rather long time for it to become a sphericalSedov-von Neumann-Taylor blast wave. We find that the late-time afterglowsbecome increasingly flatter over time. But we disagree with the common notionthat there is a sudden flattening in lightcurves due to the transition into theSedov-von Neumann-Taylor solution. We have also found that there is a bump inlightcurves at very late times ($\sim 1000$ days) due to radiation from thecounter jet. We speculate that such a counter jet bump might have already beenobserved in GRB 980703.

The Dynamics and Afterglow Radiation of Gamma-Ray Bursts. I. Constant Density Medium