The final energy spectrum in the four cases and the initial energy spectrum plots. The thick solid line with a narrow peak at E₀ = 1.3105 keV represents the initial Maxwell energy distribution. The solid, dashed, dot-dashed, and dotted extended curves indicate the simulated particles’ energy spectral distribution, averaged over the entire downstream region, at the end of the simulations (T_max = 2400), corresponding to Cases A, B, C, and D, respectively. Most particles cross the shock only once, producing the large broad peak centered at E_A ~ 0.05 keV, E_B ~ 0. 1 keV, E_C ~ 0.15 keV, and E_D ~ 0.20 keV in Cases A, B, C, and D, respectively. However, some particles gain enough energy via the Fermi acceleration mechanism to produce the “power-law” tail in the energy spectrum with the cutoff at E_A = 1.10 MeV, E_B = 2.41 MeV, E_C = 2.98 MeV, and E_D = 4.01 MeV corresponding to Cases A, B, C, and D. All these energy spectra are plotted in the same shock frame.