Calving is a major means of ice discharge of the Antarctic and Greenland Ice Sheets. The breaking off of icebergs changes the ice front configuration of marine terminating glaciers, which affects the stress regime of their upstream areas. Recent observations show the close correlation between the ice front position and the behaviour of many outlet glaciers. However, modelling of a glacier subject to calving poses various challenges. No universal calving rate parametrisation is known, and tracking of a moving ice front and the related boundary conditions in two or three spatial dimensions is non-trivial. Here, we present the theoretical and technical framework for a Level-Set Method, an implicit boundary tracking scheme, which we implemented into the Ice Sheet System Model (ISSM). The scheme allows us to study the dynamic response of a drainage basin to user-defined front ablation rates. We apply the method in a suite of experiments to Jakobshavn Isbræ, a major marine terminating outlet glacier of the western Greenland Ice Sheet. The model robustly reproduces the high sensitivity of the glacier to frontal ablation in form of calving. We find that enhanced calving is able to trigger significant acceleration of the ice stream. Upstream acceleration is sustained through a combination of various feedback mechanisms. However, lateral stress and ice influx into the trough are able to stabilise the ice stream. This study contributes to the present discussion on causes and effects of the continued changes occurring at Jakobshavn Isbræ, and emphasises that the incorporation of seasonal calving and dynamic lateral effects is key for realistic model projections of future global sea level rise on centennial time scales.