Although the mathematics of ice sheet dynamics is well-established, the prediction of profiles and grounding movement are analytically infeasible and numerically difficult to achieve. Here we present a new ‘moving-mesh’ approach to simulating ice sheets and glaciers computationally that is driven by ice diffusion movement and successfully reproduces the features of ice-flow, including advance and retreat. The method is applied to a shallow ice model of a glacier using data from one of the EISMINT test cases and the results are compared. We then show how assimilation of measured data can be used within the moving framework to improve the prediction of ice sheet movement. We develop a procedure for treating the mesh point positions, together with the ice thickness, as unknown state variables within the assimilation system. The correlation between the unknown mesh positions and the ice thickness is approximated by a simple correlation function that provides flow dependent co-variances. We demonstrate the success of the technique for noisy, very infrequent, partial measurements of ice thickness, both with and without noisy measurements of the position of the flow boundary.