The atmospheric aerosol dynamics model (AADM) has been widely used in both comprehensive air quality model systems and chemical transport modeling from road to global scales. The AADM consists of the Smoluchowski coagulation equation (SCE) which describes the atmospheric aerosol size growth due to coagulation. The numerical solution to the SCE undergoing Brownian coagulation in the free molecular regime is a direct challenge because of a stumbling block for the kernel to be expressed by an equivalent linear expression and a predefined lognormal size distribution, which is inconsistent with aerosols having bimodal or multimodal size distribution. Thus, a new mathematical method for solving the SCE without the strong assumption of log-normal size distribution is proposed and developed. This method is verified with a referenced sectional method (SM) with excellent agreement. The accuracy of the method approaches closely to the TEMOM, but overcomes the limitation of the classical log MOM. The computational time of this scheme is largely reduced when comparing to the SM. The new method is successfully implemented to reveal the formation and growth of secondary particles emitted from the vehicle exhaust tailpipe. It is surprisingly found that the formation of new particles only appears in the interface region of the turbulent exhaust jet which is very close to the tailpipe exit, while there is no new particle formation in the strong mixture along the downstream. The new method is finally verified to be an efficient and reliable numerical scheme for studying atmospheric aerosol dynamics.