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.

翻译：浅冰近似（SIA）模型通常以强形式表述，用于推断冰盖和冰川的动态特性。该模型描述了接地冰盖中非牛顿、粘性、重力驱动的冰流。SIA模型的解是速度场的闭式表达式。其缺点在于，当利用SIA速度推进冰面随时间演化时，时间步长限制与水平网格尺寸呈二次方缩放关系。本文采用弱形式表述SIA模型，并加入自由表面稳定算法（FSSA）项。数值实验发现，时间步长限制的缩放关系从二次方改进为线性，但仅适用于较大的水平网格尺寸。随后，我们通过添加原本被忽略的法向应力项扩展了弱形式公式，这使得在整个水平网格尺寸范围内均可实现线性时间步长限制，从而获得计算效率更高的SIA模型。为支撑数值结果，我们从理论上证明FSSA稳定项的加入将二阶导数表面项的显式时间步进处理切换为隐式时间步进处理。此外，我们进行了计算成本分析，结合稳定性特性和精度的数值结果，表明弱形式SIA模型优于标准SIA模型。