Increasing stable time-step sizes of the free-surface problem arising in ice-sheet simulations

Increasing numerical stability of mountain valley glacier simulations: implementation and testing of free-surface stabilization in Elmer/Ice

“…The free-surface equation is then solved for a new height function, z s (x, y, t+∆t), which in turn determines the new domain Ω(t+∆t). The mesh is updated based on an extruded-mesh principle, wherein nodes are vertically aligned in columns such that the mesh can be updated by simply displacing nodes vertically according to the new height function z s (x, y, t + ∆t); see e.g., Löfgren et al (2022) for implementation details. This process is repeated until the final simulation time is reached.…”

“…It has been demonstrated for lower-order physics models, such as SIA, that when coupled to the free-surface equation governing the evolution of ice sheets and glaciers they suffer from a parabolic type time-step size constraint that is highly dependent on the ice-domain thickness (Bueler et al, 2005;Gong et al, 2017;Bueler, 2022;Robinson et al, 2022). However, for the Stokes equations the same parabolic time-step size restriction does not necessarily hold true -even for setups where the SIA and the Stokes equations gives qualitatively similar solutions (Löfgren et al, 2022). Still, for ice-sheet simulations using the Stokes equations, the time-step sizes complying with the stability restrictions are typically found to be in the order of 0.1 to 10 yr (Gong et al, 2017;Löfgren et al, 2022).…”

“…However, for the Stokes equations the same parabolic time-step size restriction does not necessarily hold true -even for setups where the SIA and the Stokes equations gives qualitatively similar solutions (Löfgren et al, 2022). Still, for ice-sheet simulations using the Stokes equations, the time-step sizes complying with the stability restrictions are typically found to be in the order of 0.1 to 10 yr (Gong et al, 2017;Löfgren et al, 2022). This is considerably smaller than typical time-scales at which ice-sheets evolve (Hindmarsh and Payne, 1996), which can be as large as 10000 yr (Greve and Blatter, 2009).…”

“…Still, the values of the physical parameters describing ice flow are different from that in mantle convection, and aspect ratios of ice sheets can be as small as 1:1000. These issues were addressed by Löfgren et al (2022), where the FSSA method was adapted to ice-flow modeling. It was concluded that the method works well in an ice-dynamical setting, and for the problems presented showed the potential to increase the LST by an order of magnitude.…”

Performance analysis of high-resolution ice-sheet simulations

“…Stability conditions of explicit SIA models appeared some time ago (e.g. Hindmarsh and Payne, 1996), but recent studies have focussed on the stability conditions of explicit hybrid, higher-order and Stokes dynamics models (Cheng and others, 2017; Robinson and others, 2022), or on lengthening their steps (Löfgren and others, 2022).…”