The micro-channel combustion increasingly plays a significant role in Micro-Electro-Mechanical System (MEMS). The numerical investigation has been carried out in a micro-channel with detailed reaction mechanism. Also, an effective method has been identified to improve methane conversion efficiency. Moreover, the extinction limit of premixed methane/air catalytic combustion has been determined with the presence of a convex cavity. And, the effects of four factors for normalized cavity length, normalized cavity diameter, position, and number of convex cavities on the catalytic combustion characteristics of methane/air in a micro-channel have been investigated with numerical method, respectively. Comparison of the above four factors, increasing the normalized cavity diameter of the convex cavity is the most effective way to improve methane conversion ratio and expand extinction limit. The results also show that a vortex street in the channel appears with normalized cavity diameter of 0.3, the recirculation zone and vortex street zone in the channel improve the stability of combustion. Besides, the methane conversion ratio can be increased to 90.3% and the extinction limit can be reached up to 18.9 m/s at normalized cavity diameter of 0.3. This study provides some useful information for the design and operation of the micro-combustor with convex cavity.