Evolution of the total number of particles N_e with time for the same case shown in Fig. 5. Different types of lines show N_e as derived from the numerical solutions for different choices of the time step (see inset legend). The black lines represent the number of particles calculated by considering all grid points, while the blue lines represent the integration of the particle distribution within the analytically expected range of Lorentz factors, i.e. γ_e,min(t) and γ_e,max(t). For comparison, the analytical solution yields N_e,tot = 4.9 × 10⁵⁰ electrons.