Changing climate conditions on Greenland influence the snow accumulation rate and surface mass balance (SMB) on the ice sheet and, ultimately, its shape. This can in turn affect local climate via orography and albedo variations and, potentially, remote areas via changes in ocean circulation triggered by melt water or calving from the ice sheet. Examining these interactions in the IPSL global model requires improving the representation of snow at the ice sheet surface. In this paper, we present a new snow scheme implemented in LMDZ, the atmospheric component of the IPSL coupled model. We analyse surface climate and SMB on the Greenland ice sheet under insolation and oceanic boundary conditions for modern, but also for two different past climates, the last glacial inception (115 kyr BP) and the Eemian (126 kyr BP). While being limited by the low resolution of the general circulation model (GCM), present-day SMB is on the same order of magnitude as recent regional model findings. It is affected by a moist bias of the GCM in Western Greenland and a dry bias in the north-east. Under Eemian conditions, the SMB decreases largely, and melting affects areas in which the ice sheet surface is today at high altitude, including recent ice core drilling sites as NEEM. In contrast, glacial inception conditions lead to a higher mass balance overall due to the reduced melting in the colder summer climate. Compared to the widely applied positive degree-day (PDD) parameterization of SMB, our direct modelling results suggest a weaker sensitivity of SMB to changing climatic forcing. For the Eemian climate, our model simulations using interannually varying monthly mean forcings for the ocean surface temperature and sea ice cover lead to significantly higher SMB in southern Greenland compared to simulations forced with climatological monthly means.