Numerical convergence of viscous‐plastic sea ice models

Abstract

[1] We investigate the convergence properties of the nonlinear solver used in viscousplastic (VP) sea ice models. More specifically, we study the nonlinear solver that is based on an implicit solution of the linearized system of equations and an outer loop (OL) iteration (or pseudo time steps). When the time step is comparable to the forcing time scale, a small number of OL iterations leads to errors in the simulated velocity field that are of the same order of magnitude as the mean drift. The slow convergence is an issue at all spatial resolution but is more severe as the grid is refined. The metrics used by the sea ice modeling community to assess convergence are misleading. Indeed, when performing 10 OL iterations with a 6 h time step, the average kinetic energy of the pack is always within 2% of the fully converged value. However, the errors on the drift are of the same order of magnitude as the mean drift. Also, while 40 OL iterations provide a VP solution (with stress states inside or on the yield curve), large parts of the domain are characterized by errors of 0.5-1.0 cm s À1 . The largest errors are localized in regions of large sea ice deformations where strong ice interactions are present. To resolve those deformations accurately, we find that more than 100 OL iterations are required. To obtain a continuously differentiable momentum equation, we replace the formulation of the viscous coefficients with capping with a tangent hyperbolic function. This reduces the number of OL iterations required to reach a certain residual norm by a factor of $2.

Implementation of the Jacobian-free Newton–Krylov method for solving the first-order ice sheet momentum balance

“…However, as has been shown in previous work [31,32], a Picard treatment of the nonlinearity generally leads to undesirably slow rates of convergence. This paper describes the implementation of a more computationally efficient nonlinear solver, the Jacobian-Free NewtonKrylov (JFNK, [33]) method, into Glimmer-CISM.…”

Improving the numerical convergence of viscous-plastic sea ice models with the Jacobian-free Newton–Krylov method

“…In a recent paper, Lemieux and Tremblay [6] investigated the numerical convergence of VP models. They showed that the numerical convergence of the VP model standard solver is very slow, i.e., the errors on the calculated approximate velocity field decrease very slowly with the number of OL iterations.…”

“…This slow convergence is an issue at all spatial resolution but gets more severe as the grid is refined. Lemieux and Tremblay [6] pointed out that the sea ice momentum equation with a VP formulation is intrinsically difficult to solve because it is highly nonlinear. Indeed, sea ice has very little tensile strength while it can resist large stresses before yielding in compression.…”

The elastic–viscous–plastic method revisited

“…Even with 40 outer-loop iterations, the velocity field can have significant errors (as shown by Lemieux and Tremblay (2009)). A solver (e.g., GMRES or LSOR) is called at each iteration to solve the linear system of equations.…”

“…In all cases, a linear solver is needed, making the parallelization of the code more complex. These methods usually converge but with poor rates of convergence (Lemieux and Tremblay, 2009). …”