The unsteady plane flow of ice-sheets: A parabolic problem with two moving boundaries

Abstract

Abstract 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. Crank-Nicholson schemes for the solution of the equations are constructed, and iterative schemes for the solution of the resulting non-linear equations are considered. Boundary (margin) motion is governed by the surface slope at the margin. Differentiation of the evolution equations results in an evolution equation for the margin slopes. It is shown that incorporation of this evolution equation, while not formally increasing the accuracy of the finite difference schemes, in practice increases accuracy of the solution.