Acceleration of convergence of Fourier-Bessel series for piecewise smooth functions on the base of asymptotic representation of its coefficients is presented. In the step A "jumps" of the unknown function is calculated. In the step B an acceleration scheme is constructed and resulting AB algorithm is an analogue of the known "Bernolli" method for acceleration of convergence of Fourier series. In the step C Pade approximations are applied and an analogue of the "quasipolinomial" acceleration of the paper [8] is considered. In numerical results the unconditional advantages of the AC algorithm are demonstrated.