zbMATH — the first resource for mathematics

Implementation of Newton’s method with an analytical Jacobian to solve the 1D sea ice momentum equation. (English) Zbl 1380.65202
Summary: New numerical solvers are being considered in response to the rising computational cost of properly solving the sea ice momentum equation at high resolution. The Jacobian free version of Newton’s method has allowed models to obtain the converged solution faster than other implicit solvers used previously. To further improve on this recent development, the analytical Jacobian of the 1D sea ice momentum equation is derived and used inside Newton’s method. The results are promising in terms of computational efficiency. Although robustness remains an issue for some test cases, it is improved compared to the Jacobian free approach. In order to make use of the strong points of both the new and Jacobian free methods, a hybrid preconditioner using the Picard and Jacobian matrices to improve global and local convergence, respectively, is also introduced. This preconditioner combines the robustness and computational efficiency of the previously used preconditioning matrices when solving the sea ice momentum equation.
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
35Q86 PDEs in connection with geophysics
Full Text: DOI
[1] Post, E.; Bhatt, U. S.; Bitz, C. M.; Brodie, J. F.; Fulton, T. L.; Hebblewhite, M.; Kerby, J.; Kutz, S. J.; Stirling, I.; Walker, D. A., Ecological consequences of sea-ice decline, Science, 341, 519-524, (2013)
[2] Francis, J. A.; Vavrus, S. J., Evidence linking arctic amplification to extreme weather in mid-latitudes, Geophys. Res. Lett., 39, (2012)
[3] Hibler, W., A dynamic thermodynamic sea ice model, J. Phys. Oceanogr., 9, 815-846, (1979)
[4] Coon, M.; Maykut, G.; Pritchard, R., Modeling the pack ice as an elastic-plastic material, A.I.D.J.E.X. Bull., 24, 1-105, (1974)
[5] Tremblay, L.; Mysak, L., Modeling sea ice as a granular material, including the dilatancy effect, J. Phys. Oceanogr., 27, 2342-2360, (1997)
[6] Schreyer, H.; Sulsky, D.; Munday, L.; Coon, M.; Kwok, R., Elastic-decohesive constitutive model for sea ice, J. Geophys. Res., Oceans, 111, (2006)
[7] 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)
[8] Tsamados, M.; Feltham, D. L.; Wilchinsky, A. V., Impact of a new anisotropic rheology on simulations of arctic sea ice, J. Geophys. Res., Oceans, 118, 91-107, (2013)
[9] Rampal, P.; Bouillon, S.; Ólason, E.; Morlighem, M., Nextsim: a new Lagrangian sea ice model, Cryosphere, 10, 1055-1073, (2016)
[10] 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., Oceans, 105, 11299-11320, (2000)
[11] Ip, C. F.; Hibler, W. D.; Flato, G. M., On the effect of rheology on seasonal sea-ice simulations, Ann. Glaciol., 15, 17-25, (1991)
[12] Zhang, J.; Hibler, W., On an efficient numerical method for modeling sea ice dynamics, J. Geophys. Res., Oceans, 102, 8691-8702, (1997)
[13] 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., Oceans (1978-2012), 113, 12, (2008)
[14] Lemieux, J.-F.; Tremblay, B., Numerical convergence of viscous-plastic sea ice models, J. Geophys. Res., Oceans (1978-2012), 114, 14, (2009)
[15] Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 461-469, (1993) · Zbl 0780.65022
[16] 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
[17] 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
[18] 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
[19] Lemieux, J.-F.; Knoll, D.; Losch, M.; Girard, C., A second-order accurate in time implicit-explicit (IMEX) integration scheme for sea ice dynamics, J. Comput. Phys., 263, 375-392, (2014) · Zbl 1349.86006
[20] J. Williams, L.B. Tremblay, J.-F. Lemieux, Numerically resolving the propagation of plastic deformation in viscous-plastic sea-ice model, J. Comput. Phys., in press.
[21] Hunke, E.; Dukowicz, J., An elastic-viscous-plastic model for sea ice dynamics, J. Phys. Oceanogr., 27, 1849-1867, (1997)
[22] Hunke, E. C., Viscous-plastic sea ice dynamics with the EVP model: linearization issues, J. Comput. Phys., 170, 18-38, (2001) · Zbl 1030.74032
[23] 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)
[24] König Beatty, C.; Holland, D. M., Modeling landfast sea ice by adding tensile strength, J. Phys. Oceanogr., 40, 185-198, (2010)
[25] Losch, M.; Danilov, S., On solving the momentum equations of dynamic sea ice models with implicit solvers and the elastic-viscous-plastic technique, Ocean Model., 41, 42-52, (2012)
[26] 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)
[27] Bouillon, S.; Fichefet, T.; Legat, V.; Madec, G., The elastic-viscous-plastic method revisited, Ocean Model., 71, 2-12, (2013)
[28] Kimmritz, M.; Danilov, S.; Losch, M., On the convergence of the modified elastic-viscous-plastic method for solving the sea ice momentum equation, J. Comput. Phys., 296, 90-100, (2015) · Zbl 1352.86021
[29] Kimmritz, M.; Danilov, S.; Losch, M., The adaptive EVP method for solving the sea ice momentum equation, Ocean Model., 101, 59-67, (2016)
[30] Lipscomb, W. H.; Hunke, E. C.; Maslowski, W.; Jakacki, J., Ridging, strength, and stability in high-resolution sea ice models, J. Geophys. Res., 112, 18, (2007)
[31] McPhee, M., Ice-Ocean momentum transfer for the AIDJEX ice model, A.I.D.J.E.X. Bull., 29, 93-111, (1975)
[32] Arakawa, A.; Lamb, V. R., Computational design of the basic dynamical processes of the ucla general circulation model, Methods Comput. Phys., 17, 173-265, (1977)
[33] 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
[34] Godoy, W. F.; Liu, X., Parallel Jacobian-free Newton Krylov solution of the discrete ordinates method with flux limiters for 3d radiative transfer, J. Comput. Phys., 231, 4257-4278, (2012) · Zbl 1259.78048
[35] 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)
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.