Finite difference algorithms have been developed to solve a one-dimensional non-linear parabolic equation with one or two moving boundaries and to analyse the unsteady plane flow of ice-sheets. They are designed to investigate the response of an ice-sheet to changes in climate, and to reconstruct climatic changes implied by past ice-sheet variations inferred from glacial geological data. Two algorithms are presented and compared. The first, a fixed domain method, replaces time as an independent variable with span. The grid interval in real space is kept constant, and thus the number of grid points changes with span. The second, a moving mesh method, retains time as one of the independent variables, but normalises the spatial variable relative to the span, which now enters the diffusion and advection coeficients in the parabolic equation for the surface profile.