Owing to its efficiency and aptitude for massive parallelization, the lattice Boltzmann method generally outperforms conventional solvers in terms of execution time in weakly compressible flows. However, in the incompressible limit, the authorized time-step (being inversely proportional to the speed of sound) becomes prohibitively small so that the performance advantage over continuum-based solvers vanishes. A remedy to optimize the time-step is provided by tailoring an artificial speed of sound, which can be fixed or variable throughout the simulation, the latter case referring to an adaptive time-stepping. While achieving considerable speed-ups in certain flow configurations, adaptive time-stepping comes with the flaw that the continuities of mass density and pressure cannot be fulfilled conjointly when thespeed of sound is varied. Therefore, a trade-off is needed. By leaving the mass density unchanged, the conservation of mass is preserved but the pressure presents a discontinuity in the momentum equation. In contrast, manipulation of the mass density allows us to ensure the continuity of the pressure term in the momentum equation (per unit mass) but the conservation of mass is locally sacrificed. This algorithm requires an additional scaling operation and will therefore be called adaptive time-stepping with correction in the article. Interestingly, we found that this second trade-off is generally preferable.