Abstract: Wake-induced vibration (WIV) of a two-degree-of-freedom downstream square cylinder behind a stationary upstream square one are numerically investigated at low Reynolds numbers by using semi-implicit Characteristics-based split(CBS) finite element algorithm in this study. Firstly, the results are compared with the existing data in the literature to verify the accurateness of the method. Then, the influence of Reynolds number (Re) and reduced velocity (U_\rm r)on the wake-induced vibration response of the downstream square cylinder is analyzed. Meanwhile, the results are also compared with those of the single square cylinder case. The numerical results show that the Reand U_\rm rhave great influence on the dynamic response characteristics, such as amplitude, vibration frequency and motion trajectory of the downstream square cylinder. With the increasing of Re, the interaction between both cylinders changes from the vortex-induced interference to the wake one, resulting in the change of the frequency characteristics and the intensifying dynamic response of the downstream circular cylinder. The trajectories resemble figure "8" in the single square cylinder case. However, in the case of both cylinders arranged in-line, the trajectories of the downstream cylinder become complicated with the increasing of Re. The motion orbits are basically "8" figure at Re=40, 80. When the Reare 120, 160 and 200, the "dual-8" figure can be observed. Meanwhile, the wake field characteristics of the downstream square cylinder show 2S, 2S*, 2P, 2T, P+S and steady mode. Finally, the wake-induced vibration mechanisms of two square cylinders in the tandem arrangement is revealed according to the analysis of the characteristic of the flow field.