A system compensates for heave induced pressure fluctuations on a floating rig when a drill string or tubular is lifted off bottom and suspended on the rig, such as when tubular connections are made during MPD, tripping, or when a kick is circulated out during conventional drilling. In one embodiment, a liquid and a gas interface moves along a flow line between a riser and a gas accumulator as the tubular moves up and down. In another embodiment, a pressure relief valve or adjustable choke allows the movement of fluid from the riser when the tubular moves down, and a pump with a pressure regulator moves fluid to the riser when the tubular moves up. In other embodiments, a piston connected with the rig or the riser telescoping joint moves in a fluid container thereby communicating a required amount of the fluid either into or out of the riser annulus. The system also compensates for heave induced pressure fluctuations on a floating rig when a riser telescoping joint located below a RCD is moving while drilling.