The process of formation or transformation of multi-armed spiral patterns in gas discharge system is investigated numerically by using H.-G. Purwins model with three components. The parameter space is obtained though analyzing the influence of system parameters on system space, where a stable spiral appears. Besides, the formation mechanism and spatiotemporal characteristics of spiral pattern are studied. In addition, the evolution process of pattern from simple hexagon to spiral wave is numerically simulated, and various kinds of spirals are obtained (including two-armed, three-armed, four-armed, five-armed, six-armed, and seven-armed spirals). It is found that the stable spiral only survives in Turing-Hopf space, which is the result of interaction between Turing mode and Hopf mode. Furthermore, the spiral tips constantly rotate for various spiral patterns, and the velocity increases with the number of spiral arm increasing. For the influences of perturbation and boundary conditions, the multi-armed spiral pattern can lose one arm and become a new spiral in the rotating process. In conclusion, the numerical simulation results are in good agreement with those obtained in gas discharge experiment.