The Shallow Ice Approximation (SIA) model on strong form is commonly used for inferring the flow dynamics of grounded ice sheets. The solution to the SIA model is a closed-form expression for the velocity field. When that velocity field is used to advance the ice surface in time, the time steps have to take small values due to 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 form by adding the initially neglected normal stress terms. This allows for a linear time step restriction across the whole range of the horizontal mesh sizes, leading to an improved efficiency. Theoretical analysis demonstrates that the inclusion of FSSA stabilization terms transitions the explicit time stepping treatment of second derivative surface terms to an implicit approach. Moreover, a computational cost analysis, combined with numerical results on stability and accuracy, advocates for preferring the SIA models written on weak form over the standard SIA model.

翻译：浅冰近似模型（SIA）通常基于强形式，用于推断接地冰盖的流动动力学。SIA模型的解是速度场的闭式表达式。当利用该速度场随时间推进冰面时，由于时间步长与水平网格尺寸呈二次标度关系，因此必须采用较小的步长值。本文提出将SIA模型写成弱形式，并引入自由表面稳定化算法（FSSA）项。数值实验发现，时间步长限制的标度关系从二次改进为线性，但仅适用于较大的水平网格尺寸。随后，通过添加最初忽略的正应力项扩展弱形式，使得在整个水平网格尺寸范围内均能实现线性时间步长限制，从而提升计算效率。理论分析表明，FSSA稳定化项的加入将二阶导数表面项的显式时间推进处理转化为隐式方法。此外，结合稳定性与精度的数值结果进行的计算成本分析，支持优先采用弱形式SIA模型而非标准SIA模型。