Pumping for chromatograph applications calls for constant flow at high pressures and for accurate control of liquid composition. A two-cylinder pump, and associated controls provide an accurate, fixed flow rate. During the initial portion of the pumping stroke of a first piston, while its check valve remains closed, a second piston provides all the flow required at a fixed rotational speed. Once the pressure inside the first piston's cylinder has reached that of the discharge and its discharge check valve opens, the control system switches to a constant-pressure control mode, to maintain a fixed flow rate. Constant-pressure control is switched off when the second piston's discharge check valve closes, and pump control returns to fixed rotational speed. Determining the angular displacement at which to effect these control-mode switches is carried out by monitoring changes in the discharge pressure.