A second-order implicit time-stepping scheme for full-Stokes ice-sheet models

We present a second-order implicit method for coupling the free surface and velocity in full-Stokes ice-sheet models [1]. The method enables the use of very large time steps, in contrast to standard implicit solvers, which diverge in this regime.

Convergence of the implicit solver is achieved by adding and subtracting two so-called FSSA (Free Surface Stabilization Approach) stabilization terms. Unlike previous FSSA formulations, the stabilization terms do not affect the accuracy of the final solution, as they vanish upon convergence. This stabilized implicit iteration scheme is then used to construct a second-order backward differentiation formula (BDF2) time-integration method.

The method is implemented and evaluated on idealized and realistic ice-sheet test cases in Elmer/Ice and Biceps, demonstrating improved stability, accuracy, and computational efficiency. While developed for glaciological applications, the approach is easy to implement and directly applicable to other viscous free-surface Stokes problems.

 

[1] Ahlkrona, J., Henry, A. C. J., and Löfgren, A.: A fully implicit second-order method for viscous free-surface Stokes flow – application to glacier simulations, EGUsphere, 2025, 1–24, https://doi.org/10.5194/egusphere-2025-4359