Laser wake field acceleration (LWFA) is limited by some determinative aspects, such as wave breaking, dephasing, pulse divergence, etc. One of the proposed methods to overcome the acceleration limitations of the LWFA is spatio-temporal controlling of the laser focus, named flying focus. In this article, flying focus dynamics for two Bessel beam profiles is derived with complex source point method (CSPM). We investigate the two pulse with basic Gaussian and Donut-like profiles, those focal points move at a speed very close to the speed of light. Our selected flying focus parameters maintain laser intensity up to 16 times the Rayleigh length (− 8ZR, + 8ZR). We also examined the flying focus LWFA (FF-LWFA) energy gain and trapping rate for these two cases. Numerical results show that accelerated electron bunch is converged considerably for the Donut shape profile. This convergence is due to the inward radial ponderomotive force, and furthermore suitable phase difference between longitudinal and transverse wake field components. Since electrons are hold near the axis for a longer distance and keep more energy from the wake. In average 2.8% of the injected electrons are accelerated up to 255 MeV for the donut-like profiles, while only 1.4% of electrons are accelerated up to 155 MeV for Gaussian profile
