The Lorentz factor (Γ) is an important parameter related to the relativistic jet physics. We study the evolution patterns of Γ within gamma-ray burst (GRB) and active galactic nuclear jets for individual GRB 090168, GRB 140508A, and 3C 454.3. By estimating the Γ values for well-separated pulses in GRBs 090618 and 140508A with an empirical relation derived from typical GRBs, we find that the Γ evolution pattern in the two GRBs are different. The increasing-to-coasting evolution pattern of Γ in GRB 090618 likely indicates that the GRB fireball is still being accelerated in the prompt phase. The clear decrease evolution pattern of Γ in GRB 140508A suggests the deceleration of the fireball components. By deriving the Γ value through fitting their spectral energy distribution in different flares of 3C 454.3, a pattern of Γ-tracking-γ-ray flux is clearly found, likely indicating that the observed gamma-ray flares are being due to the Doppler boosting effect to the jet emission.

Schwarz waveform relaxation (SWR) algorithm has been investigated deeply and widely for regular time dependent problems. But for time delay problems, complete analysis of the algorithm is rare. In this paper, by using the reaction diffusion equations with a constant discrete delay as the underlying model problem, we investigate the convergence behavior of the overlapping SWR algorithm with Robin transmission condition. The key point of using this transmission condition is to determine a free parameter as better as possible and it is shown that the best choice of the parameter is determined by the solution of a min-max problem, which is more complex than the one arising for regular problems without delay. We propose new notion to solve the min-max problem and obtain a quasi-optimized choice of the parameter, which is shown efficient to accelerate the convergence of the SWR algorithm. Numerical results are provided to validate the theoretical conclusions.