A new algorithm, the modified direct simulation Monte-Carlo (MDSMC) method, for the simulation of Couette-Taylor gas flow problem is developed. The Taylor series expansion is used to obtain the modified equation of the first-order time discretization of the collision equation and the new algorithm, MDSMC, is implemented to simulate the collision equation in the Boltzmann equation. In the new algorithm (MDSMC) there exists a new extra term which takes in to account the effect of the second order collision. This new extra term has the effect of enhancing the appearance of the first Taylor instabilities of vortices streamlines. In the new algorithm (MDSMC) there also exists a second order term in time step in the probabilistic coefficients which has the effect of simulation with higher accuracy than the previous DSMC algorithm. The appearance of the first Taylor instabilities of vortices streamlines using the MDSMC algorithm at different ratios of $omega/ u$ (experimental data of Taylor [1]) occurred at less time-step than using the DSMC algorithm. The results of the torque developed on the stationary cylinder using the MDSMC algorithm show better agreement in comparison with the experimental data of Kuhlthau [2] than the results of the torque developed on the stationary cylinder using the DSMC algorithm.