Accurate and stable time stepping in ice sheet modeling

TL;DR

This paper presents an adaptive time stepping method for ice sheet simulations that balances stability and accuracy efficiently, using predictor-corrector pairs and error estimation to optimize time step size.

Contribution

The paper introduces a novel adaptive time stepping approach for ice sheet modeling that reduces computational effort while maintaining stability and accuracy.

Findings

The method effectively balances stability and accuracy in simulations.

The adaptive time step is optimized based on error estimates and stability constraints.

Experimental results align well with theoretical stability bounds.

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.