A second-order accurate in time implicit-explicit (IMEX) integration scheme for sea ice dynamics. (English) Zbl 1349.86006

Summary: Current sea ice models use numerical schemes based on a splitting in time between the momentum and continuity equations. Because the ice strength is explicit when solving the momentum equation, this can create unrealistic ice stress gradients when using a large time step. As a consequence, noise develops in the numerical solution and these models can even become numerically unstable at high resolution. To resolve this issue, we have implemented an iterated IMplicit-EXplicit (IMEX) time integration method. This IMEX method was developed in the framework of an already implemented Jacobian-free Newton-Krylov solver. The basic idea of this IMEX approach is to move the explicit calculation of the sea ice thickness and concentration inside the Newton loop such that these tracers evolve during the implicit integration. To obtain second-order accuracy in time, we have also modified the explicit time integration to a second-order Runge-Kutta approach and by introducing a second-order backward difference method for the implicit integration of the momentum equation. These modifications to the code are minor and straightforward. By comparing results with a reference solution obtained with a very small time step, it is shown that the approximate solution is second-order accurate in time. The new method permits to obtain the same accuracy as the splitting in time but by using a time step that is 10 times larger. Results show that the second-order scheme is more than five times more computationally efficient than the splitting in time approach for an equivalent level of error.


86-08 Computational methods for problems pertaining to geophysics
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
86A60 Geological problems


MITgcm; Trilinos
Full Text: DOI


[1] Hibler, W. D., A dynamic thermodynamic sea ice model, J. Phys. Oceanogr., 9, 815-846, (1979)
[2] Lipscomb, W. H.; Hunke, E. C.; Maslowski, W.; Jakacki, J., Ridging, strength, and stability in high-resolution sea ice models, J. Geophys. Res., 112, (2007)
[3] Tremblay, B.; Mysak, L. A., Modeling sea ice as a granular material, including the dilatancy effect, J. Phys. Oceanogr., 27, 2342-2360, (1997)
[4] Lemieux, J.-F.; Tremblay, B.; Sedláček, J.; Tupper, P.; Thomas, S.; Huard, D.; Auclair, J.-P., Improving the numerical convergence of viscous-plastic sea ice models with the Jacobian-free Newton Krylov method, J. Comput. Phys., 229, 2840-2852, (2010) · Zbl 1184.86004
[5] Losch, M.; Menemenlis, D.; Campin, J.-M.; Heimbach, P.; Hill, C., On the formulation of sea-ice models. part 1: effects of different solver implementations and parameterizations, Ocean Model., 33, 129-144, (2010)
[6] Hutchings, J. K.; Jasak, H.; Laxon, S. W., A strength implicit correction scheme for the viscous-plastic sea ice model, Ocean Model., 7, 111-133, (2004)
[7] Zhang, J.; Hibler, W. D., On an efficient numerical method for modeling sea ice dynamics, J. Geophys. Res., 102, 8691-8702, (1997)
[8] Lemieux, J.-F.; Tremblay, B., Numerical convergence of viscous-plastic sea ice models, J. Geophys. Res., 114, C05009, (2009)
[9] Hunke, E. C., Viscous-plastic sea ice dynamics with the EVP model: linearization issues, J. Comput. Phys., 170, 18-38, (2001) · Zbl 1030.74032
[10] Bouillon, S.; Fichefet, T.; Legat, V.; Madec, G., The elastic-viscous-plastic method revisited, Ocean Model., 71, 2-12, (2013)
[11] Lemieux, J.-F.; Knoll, D. A.; Tremblay, B.; Holland, D. M.; Losch, M., A comparison of the Jacobian-free Newton Krylov method and the EVP model for solving the sea ice momentum equation with a viscous-plastic formulation: a serial algorithm study, J. Comput. Phys., 231, 5926-5944, (2012)
[12] Losch, M.; Fuchs, A.; Lemieux, J.-F.; Vanselow, A., A parallel jacobian-free Newton-Krylov solver for a coupled sea ice-Ocean model, J. Comput. Phys., 257, 901-911, (2014) · Zbl 1349.86008
[13] MITgcm group, Mitgcm user manual, online documentation, MIT/EAPS, (2012), MIT Cambridge, MA 02139, USA, Technical Report
[14] Durran, D. R.; Blossey, P. N., Implicit-explicit multistep methods for fast-wave-slow-wave problems, Mon. Weather Rev., 140, 1307-1325, (2012)
[15] Kadioglu, S. Y.; Knoll, D. A., A fully second order implicit/explicit time integration technique for hydrodynamics plus nonlinear heat conduction problems, J. Comput. Phys., 229, 3237-3249, (2010) · Zbl 1307.76056
[16] Kadioglu, S. Y.; Knoll, D. A.; Lowrie, R. B.; Rauenzahn, R. M., A second order self-consistent IMEX method for radiation hydrodynamics, J. Comput. Phys., 229, 8313-8332, (2010) · Zbl 1381.76262
[17] Girard, L.; Weiss, J.; Molines, J. M.; Barnier, B.; Bouillon, S., Evaluation of high-resolution sea ice models on the basis of statistical and scaling properties of arctic sea ice drift and deformation, J. Geophys. Res., 114, (2009)
[18] Rampal, P.; Weiss, J.; Marsan, D.; Lindsay, R.; Stern, H., Scaling properties of sea ice deformation from buoy dispersion analysis, J. Geophys. Res., 113, (2008)
[19] Schreyer, H. L.; Sulsky, D. L.; Munday, L. B.; Coon, M. D.; Kwok, R., Elastic-decohesive constitutive model for sea ice, J. Geophys. Res., 111, (2006)
[20] Girard, L.; Bouillon, S.; Weiss, J.; Amitrano, D.; Fichefet, T.; Legat, V., A new modeling framework for sea-ice mechanics based on elasto-brittle rheology, Ann. Glaciol., 52, 123-132, (2011)
[21] Wang, K.; Wang, C., Modeling linear kinematic features in pack ice, J. Geophys. Res., 114, C12011, (2009)
[22] Coon, M. D.; Maykut, G. A.; Pritchard, R. S.; Rothrock, D. A.; Thorndike, A. S., Modeling the pack ice as an elastic-plastic material, A.I.D.J.E.X. Bull., 24, 1-105, (1974)
[23] McPhee, M., Ice-Ocean momentum transfer for the AIDJEX ice model, A.I.D.J.E.X. Bull., 29, 93-111, (1975)
[24] Kreyscher, M.; Harder, M.; Lemke, P.; Flato, G. M., Results of the sea ice model intercomparison project: evaluation of sea ice rheology schemes for use in climate simulations, J. Geophys. Res., 105, 11299-11320, (2000)
[25] Flato, G. M.; Hibler, W. D., Modeling pack ice as a cavitating fluid, J. Phys. Oceanogr., 22, 626-651, (1992)
[26] Dukowicz, J. K., Comments on the “stability of the viscous-plastic sea ice rheology”, J. Phys. Oceanogr., 27, 480-481, (1997)
[27] Lemieux, J.-F.; Tremblay, B.; Thomas, S.; Sedláček, J.; Mysak, L. A., Using the preconditioned generalized minimum residual (GMRES) method to solve the sea-ice momentum equation, J. Geophys. Res., 113, C10004, (2008)
[28] Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 461-469, (1993) · Zbl 0780.65022
[29] Knoll, D. A.; Keyes, D. E., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 357-397, (2004) · Zbl 1036.65045
[30] Eisenstat, S. C.; Walker, H. F., Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17, 16-32, (1996) · Zbl 0845.65021
[31] Kalnay, E.; Kanamitsu, M.; Kistler, R.; Collins, W.; Deaven, D.; Gandin, L.; Iredell, M.; Saha, S.; White, G.; Woollen, J.; Zhu, Y.; Leetmaa, A.; Reynolds, R.; Chelliah, M.; Ebisuzaki, W.; Higgins, W.; Janowiak, J.; Mo, K. C.; Ropelewski, C.; Wang, J.; Jenne, R.; Joseph, D., The NCEP/NCAR 40-year reanalysis project, Bull. Am. Meteorol. Soc., 77, 437-471, (1996)
[32] Lipscomb, W. H.; Hunke, E. C., Modeling sea ice transport using incremental remapping, Mon. Weather Rev., 132, 1341-1354, (2004)
[33] An, H.-B.; Bai, Z.-Z., A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations, Appl. Numer. Math., 57, 235-252, (2007) · Zbl 1123.65040
[34] Heroux, M. A.; Bartlett, R. A.; Howe, V. E.; Hoekstra, R. J.; Hu, J. J.; Kolda, T. G.; Lehoucq, R. B.; Long, K. R.; Pawlowski, R. P.; Phipps, E. T.; Salinger, A. G.; Thornquist, H. K.; Tuminaro, R. S.; Willenbring, J. M.; Williams, A., An overview of the trilinos project, ACM Trans. Math. Softw., 31, 397-423, (2005) · Zbl 1136.65354
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.