Normalized errors, ε, defined by Eq. (48), in the velocity of charges as a function of the ratio between the integration time step Δt and Δt_CFL, accumulated after running the simulation until t = t_F. Panel a: simulations of acoustic waves, and panel b: simulations of Alfvén waves. Blue triangles show results for β = 1 (defined in Eq. (21)); green circles are for β = 0.6, and red crosses are for β = 0.5. Solid lines show the results of quadratic polynomial fit ε = a(Δt/Δt_CFL)² + b(Δt/Δt_CFL)+c, with coefficients given in Tables 4 and 5.