A valveless pump consisting of a pumping chamber with an elastic tube was simulated using an immersed boundary method. The interaction between the motion of the elastic tube and the pumping chamber generated a net flow toward the outlet through a full cycle of the pump. The net flow rate of the valveless pump was examined by varying the stretching coefficient, bending coefficient, and aspect ratio of the elastic tube. Photographs of the fluid velocity vectors and the wave motions of the elastic tube were examined over one cycle of the pump to gain a better understanding of the mechanism underlying the valveless pump. The relationship between the gap in the elastic tube and the average flow rate of the pump was analyzed.