Weak form Shallow Ice Approximation models with an improved time step restriction

The Shallow Ice Approximation (SIA) model written on strong form is commonly used for inferring the dynamics of ice sheets and glaciers. The model describes non-Newtonian, viscous, and gravity driven flow of ice in grounded ice sheets. The solution to the SIA model is a closed-form expression for the velocity field. A disadvantage is that when using the SIA velocities to advance the ice surface in time, the time step restriction has a quadratic scaling in terms of the horizontal mesh size. In this paper we write the SIA model on weak form, and add in the Free Surface Stabilization Algorithm (FSSA) terms. We find numerically that the time step restriction scaling is improved from quadratic to linear, but only for large horizontal mesh sizes. We then extend the weak formulation by adding in the normal stress terms which are originally neglected. This allows for a linear time step restriction across the whole range of the horizontal mesh sizes and as such leads to a computationally more efficient SIA model. To support the numerical results we theoretically show that the addition of the FSSA stabilization terms switches the explicit time stepping treatment of the second derivative surface terms to an implicit time stepping treatment. In addition we perform a computational cost analysis, which, when combined with the numerical results on stability properties and accuracy, speaks for favouring SIA models on weak form over the standard SIA model.