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Stable finite volume element schemes for the shallow-ice approximation

  • ED BUELER (a1)

The isothermal, non-sliding shallow-ice approximation, combined with mass conservation, is a fundamental model for ice-sheet and glacier flow. It determines the ice extent, geometry and velocity by the solution of a free-boundary problem. In this paper, the steady-state form of this problem is solved directly, without time-stepping, thereby demonstrating a fully implicit scheme with no stability restrictions. The classical Mahaffy (1976) finite difference method is first reinterpreted as a ‘finite volume element’ scheme that has both an everywhere-defined approximate thickness function and an approximation of the conservation equation in flux integral form. From this reinterpretation an improved scheme is built by using better quadrature in the integral and upwinding on that part of the flux which is proportional to the bed gradient. The discrete equations are then solved by a parallel Newton scheme which respects the constraint that ice thickness is non-negative. The results show good accuracy on both flat-bed and bedrock-step exact solutions. The method is then applied at high resolution to model the steady-state geometry of the Greenland ice sheet, using only bedrock elevation and present-day surface mass balance as input data.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
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Correspondence: Ed Bueler <>
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