In loosely coupled systems and large-scale systems, Monte Carlo criticality calculation suffers from slow fission source convergence because of the high dominance ratio (DR). In previous work, the Wielandt method and the superhistory method have been separately proposed to accelerate source convergence. However, although both methods decrease the DR, they are found not able to sufficiently accelerate fission source convergence. In this paper, the effective DR is defined and used to analyze the effectiveness of the Wielandt method and the superhistory method and to theoretically prove that they cannot reduce the computational time to converge the fission source. Accordingly, both methods are modified by adjusting the source population of inactive cycles, and their efficiency after adjustment is also compared. Moreover, the asymptotic Wielandt method (AWM) and the asymptotic superhistory method (ASM) are proposed, and the rules of deciding asymptotic parameters are also discussed. The new methods are implemented into the RMC code and validated by calculating loosely coupled problems and large-scale problems. Numerical calculation results show that AWM and ASM are practical and efficient for source convergence acceleration, which can save 75% to 90% of the computational time to reach a converged fission source.