Velocity gradient convergence test for the different solvers. Top left panel: line-of-sight velocity profile used for the test. Top right panel: Associated convergence plot for the six-level Ca II setup with a Krylov relative tolerance of 10⁻² and initial LTE population densities. All the solvers required more iterations to converge than in the case shown in Fig. 6. Bottom panel: converged spectrum including the velocity gradient for RH92 (black) and JFNK (GMRES) (blue). A JFNK (GMRES) velocity-free reference spectrum (dashed black) is also shown. A blueshift clearly occurs near the line center due to the positive velocity gradient at the base of the chromosphere, resulting in very asymmetric output lines.