It follows from the local mixing length theory that the convection in the atmospheres of yellow and red giants and super-giants become supersonic. In this work we studied the physical reality of such phenomenon and its possible consequences on the structure and evolution of stars involving such situations. The main conclusion is that the supersonic nature of convection as predicted by the local mixing length theory has been overestimated. If supersonic convection is not an artifact in all situations, it is the case at least for all yellow giants and super-giants. We believe that such an artifact is due to the imperfect of the mixing length theory. Owing to the fact that convective energy transport in the atmospheres of yellow giants and super-giants is quite negligible, these artifacts have very limited consequences on the structure and evolution of these stars. However, it is not the case for red giants and super-giants whose properties can be seriously affected by this overestimation. Our investigation shows that the theoretical red phases of stars under consideration are somewhat too blue as predicted with the usual mixing length theory. To this aim, full hydrodynamic simulation is needed in order to clear the doubts on the existence of supersonic convection in these red objects.