Realistic predictions of the behaviour of marine ice sheets require that models are able to robustly simulate grounding line migration. Fixed-grid ice sheet models have been shown to exhibit inconsistent and hence unreliable grounding line migration, except at very high resolution not yet achievable in whole ice sheet simulations. In this study we present several different approaches to parameterising the grounding line. These are distinguished by choices regarding the ice thickness profile from the last grounded to the first floating grid point, and how this profile impacts the gravitational driving stress and basal drag. We demonstrate that the most obvious choice of thickness parameterisation, linear interpolation from the last grounded to the first floating grid point, is not the most effective. We show that use of a grounding line parameterisation greatly improves performance, and that choice of a better grounding line parameterisation over a simpler one can bring further improvements, in terms of both accuracy and self consistent behaviour, comparable to increasing the grid resolution by factor two (i.e. doubling the number of grid points). The approach presented here to parameterising the grounding line does not completely solve the grounding line problem, however it reduces the resolution required. The parameterisations are presented in the context of a one dimensional "shelfy-stream" flow-line model, but could be extended to cope with more than one dimension and other model formulations.