Is the distribution of the chargs on the antenna is the same

Is the distribution of the chargs on the antenna is the same as the current distribution? and why we do the mesh for the antenna during the modelling?

Ofcourse. For example, in the current direction, at the antennas end/edge the current has to go to zero while the charges pile up.
In antenna analysis & design, the unknown is the current and/or charge distribution on the antennas. In the process of using of numerical methods, we devide the antennas structure into small segments, and assume a known current and/or charge distribution over that segment and try to find the unknown amplitude of that current and/or charge distribution over that segment. For more explanation, you could, for example, refer to procedure of method of moments.