The atomic cascade in $mu^- p$ and $pi^- p$ atoms has been studied with the improved version of the extended cascade model in which new quantum mechanical calculations of the differential and integral cross sections of the elastic scattering, Stark transitions and Coulomb de-excitation have been included for the principal quantum number values $nle 8$ and the relative energies $E ge 0.01$ eV. The $X$-ray yields and kinetic energy distributions are compared with the experimental data.