Grounding line migration is a key process affecting the stability of marine ice sheets such as the West Antarctic ice sheet. Recent studies have shown that ice sheet models employing a fixed spatial grid (such as are commonly used for whole ice sheet simulations) cannot be used to solve this problem in a robust manner. We have developed a one-dimensional (vertically integrated) “shelfy stream” ice sheet model that employs the adaptive mesh refinement (AMR) technique to bring higher resolution to spatially and temporally evolving subregions of the model domain. A higher-order solver, the piecewise parabolic method (PPM), is used to compute the thickness evolution. Both AMR and PPM extend readily to greater than one dimension and could be used in full ice sheet simulations. We demonstrate that this approach can bring improvements in terms of accuracy and consistency in both grounded ice sheet and ice stream/ice shelf simulations, given the appropriate choice of refinement criteria. In particular, we demonstrate that AMR, in conjunction with a parameterization for subgrid scale grounding line position, can produce predictions of grounding line migration.