Convergence of the GMRES solver without preconditioning (left panels) and with the physics-based preconditioner (right panels). The upper panels correspond to the 2D isentropic vortex, and the lower panels to the 3D Taylor-Green vortex. In both cases, the Mach numbers considered are M_s = 10^-1, 10^-2, 10^-4, 10^-6 (the right panels assume the same legend as the left ones). The maximum allowed number of GMRES iterations was set to 300. The mean values of the number of iterations for convergence is plotted, with shaded areas showing maximum and minimum values. For each Mach number, the location of CFL_hydro corresponding to CFL_adv = 1 is shown by a vertical dashed line.