Accurate and stable time stepping in ice sheet modeling

Abstract

In this paper we introduce adaptive time step control for simulation of evolution of ice sheets. The discretization error in the approximations is estimated using "Milne's device" by comparing the result from two different methods in a predictor-corrector pair. Using a predictor-corrector pair the expensive part of the procedure, the solution of the velocity and pressure equations, is performed only once per time step and an estimate of the local error is easily obtained. The stability of the numerical solution is maintained and the accuracy is controlled by keeping the local error below a given threshold using PI-control. Depending on the threshold, the time step $\Delta t$ is bound by stability requirements or accuracy requirements. Our method takes a shorter $\Delta t$ than an implicit method but with less work in each time step and the solver is simpler. The method is analyzed theoretically with respect to stability and applied to the simulation of a 2D ice slab and a 3D circular ice sheet. %The automatically chosen $\Delta t$ is either restricted by accuracy or stability depedning on an error tolerance. The stability bounds in the experiments are explained by and agree well with the theoretical results

A comparison of the stability and performance of depth-integrated ice-dynamics solvers

“…The SSA analysis arises naturally, since the SSA equations have the same form as the DIVA solver. The numerical stability of an SIA solver under the above assumptions has been found previously (Hindmarsh, 2001;Cheng et al, 2017), showing that the maximum stable time step under the simplifications above is proportional to the square of the grid resolution:…”

“…We caution that the results depend on details of the numerical schemes and may not apply to all situations, such as when the rheology is nonlinear. Our aim is not to consider all possible schemes, or to produce a numerical scheme of optimal stability as in Cheng et al (2017), but rather to examine stability properties when applying a representative finitedifference scheme to different stress-balance equations. As in Cheng et al (2017) our analysis is applied to a linearized form of the coupled stress and continuity equations under small perturbations.…”

Description and validation of the ice-sheet model Yelmo (version 1.0)

“…These models include hybrid approaches that heuristically combine the SIA with the shallow shelf approximation (SSA) (e.g., Bueler and Brown, 2009;Winkelmann et al, 2011;Pollard and DeConto, 2012;Pattyn, 2017;Quiquet et al, 2018) and higher-order approximations (e.g., Goldberg, 2011;Cornford et al, 2013;Hoffman et al, 2018;Lipscomb et al, 2019), including full Stokes solutions (e.g., Larour et al, 2012;Gagliardini et al, 2013). Newer models often feature finite-element/finite-volume methods (e.g., Larour et al, 2012;Gagliardini et al, 2013;Hoffman et al, 2018) or adaptive mesh refinement (Cornford et al, 2013), which allows simulation of complex terrain and very high resolution where it is needed (e.g., at the grounding line in Antarctica). While more complex models are driving ad-vances in our understanding of the physics and relevant processes of ice sheets over a range of timescales, simpler and thus faster methods are still required to understand the evolution of the ice sheets on multimillennial timescales.…”

Benchmarking the vertically integrated ice-sheet model IMAU-ICE (version 2.0)

“…The mass conservation equation is integrated through time using an explicit solver; a semi-implicit solver is available, offering improved numerical stability at an increased computational expense. The model has a dynamic time-step, which is calculated using a predictor/corrector method to provide an estimate of the truncation error in the ice thickness (Cheng et al, 2016). The implementation of this method is adopted from Yelmo (Robinson et al, 2020).…”