This paper explores the energy-saving advantages of the burst-and-glide swimming and compares it with the normal self-swimming for a thunniform swimmer. The virtual swimmer allows us to perform controlled numerical experiments by varying the swinging tail number and the duty cycle while keeping the other parameters fixed. 3-D Navier-Stokes equations are used to compute the viscous flow over the swimmer. The user-defined functions and the dynamic mesh technology are used to simulate the burst-and-glide swimming. The results show that with the increase of the swinging tail number or the duty cycle, the swimming velocity, the power and the efficiency all increase, but the velocity-power ratio decreases somewhat. Therefore, choosing smaller swinging tail number and duty cycle is beneficial in reducing the power and increasing the velocity-power ratio, and thus to obtain the same velocity, less power is consumed. And to swim the same distance, the energy can significantly be saved. The power consumption, the efficiency and the velocity-power ratio in the burst-and-glide case are 43.9％, 40.6％ and 1.15 times of those in the normal swimming case, respectively. The flow structures clearly show the evolution process around the fish in the burst-and-glide swimming. The findings can be used to reasonably plan the swimming action and to take the advantage of the extal flow field energy for the fishlike robot, to be more efficient and energy-saving.