×

zbMATH — the first resource for mathematics

Improving the numerical convergence of viscous-plastic sea ice models with the Jacobian-free Newton-Krylov method. (English) Zbl 1184.86004
Summary: We have implemented the Jacobian-free Newton-Krylov (JFNK) method to solve the sea ice momentum equation with a viscous-plastic (VP) formulation. The JFNK method has many advantages: the system matrix (the Jacobian) does not need to be formed and stored, the method is parallelizable and the convergence can be nearly quadratic in the vicinity of the solution. The convergence rate of our JFNK implementation is characterized by two phases: an initial phase with slow convergence and a fast phase for which the residual norm decreases significantly from one Newton iteration to the next. Because of this fast phase, the computational gain of the JFNK method over the standard solver used in existing VP models increases with the required drop in the residual norm (termination criterion). The JFNK method is between 3 and 6.6 times faster (depending on the spatial resolution and termination criterion) than the standard solver using a preconditioned generalized minimum residual method. Resolutions tested in this study are 80, 40, 20 and 10 km. For a large required drop in the residual norm, both JFNK and standard solvers sometimes do not converge. The failure rate for both solvers increases as the grid is refined but stays relatively small (less than 2.3% of failures). With increasing spatial resolution, the velocity gradients (sea ice deformations) get more and more important. Nonlinear solvers such as the JFNK method tend to have difficulties when there are such sharp structures in the solution. This lack of robustness of both solvers is however a debatable problem as it mostly occurs for large required drops in the residual norm. Furthermore, when it occurs, it usually affects only a few grid cells, i.e., the residual is small for all the velocity components except in very localized regions. Globalization approaches for the JFNK solver, such as the line search method, have not yet proven to be successful. Further investigation is needed.

MSC:
86A05 Hydrology, hydrography, oceanography
76M25 Other numerical methods (fluid mechanics) (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Coon, M.D.; Maykut, G.A.; Pritchard, R.S.; Rothrock, D.A.; Thorndike, A.S., Modeling the pack ice as an elastic-plastic material, AIDJEX bull., 24, 1-105, (1974)
[2] Flato, G.M.; Hibler, W.D., Modeling pack ice as a cavitating fluid, J. phys. oceanogr., 22, 626-651, (1992)
[3] Hibler, W.D., A dynamic thermodynamic sea ice model, J. phys. oceanogr., 9, 815-846, (1979)
[4] Zhang, J.; Hibler, W.D., On an efficient numerical method for modeling sea ice dynamics, J. geophys. res., 102, C4, 8691-8702, (1997)
[5] 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)
[6] Lemieux, J.-F.; Tremblay, B., Numerical convergence of viscous-plastic sea ice models, J. geophys. res., 114, C05009, (2009)
[7] Eisenstat, S.C.; Walker, H.F., Globally convergent inexact Newton methods, SIAM J. optim., 4, 393-422, (1994) · Zbl 0814.65049
[8] 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
[9] Reisner, J.M.; Mousseau, V.A.; Wyszogrodzki, A.A.; Knoll, D.A., An implicitly balanced hurricane model with physics-based preconditioning, Mon. weather rev., 133, 1003-1022, (2005)
[10] Abbett, W.P., The magnetic connection between the convection zone and corona in the quiet Sun, Astrophys. J., 665, 1469-1488, (2007)
[11] Ferm, L.; Lötstedt, P., Numerical method for coupling the macro and meso scales in stochastic chemical kinetics, BIT numer. math., 47, 4, 735-762, (2007) · Zbl 1137.65006
[12] Hunke, E.C.; Dukowicz, J.K., An elastic-viscous-plastic model for sea ice dynamics, J. phys. oceanogr., 27, 1849-1867, (1997)
[13] Tremblay, B.; Mysak, L.A., Modeling sea ice as a granular material, including the dilatancy effect, J. phys. oceanogr., 27, 2342-2360, (1997)
[14] McPhee, M.G., Ice-Ocean momentum transfer for the AIDJEX ice model, AIDJEX bull., 29, 93-111, (1975)
[15] 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)
[16] Dukowicz, J., Comments on the “stability of the viscous-plastic sea ice rheology”, J. phys. oceanogr., 27, 480-481, (1997)
[17] Hibler, W.D.; Ackley, S.F., Numerical simulation of the weddell sea pack ice, J. geophys. res., 88, 2873-2887, (1983)
[18] Quarteroni, A.; Sacco, R.; Saleri, F., Numerical mathematics, (2000), Springer · Zbl 0943.65001
[19] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 3, 856-869, (1986) · Zbl 0599.65018
[20] Saad, Y., Iterative methods for sparse linear systems, (1996), PWS · Zbl 1002.65042
[21] Qin, N.; Ludlow, D.K.; Shaw, S.T., A matrix-free preconditioned Newton/GMRES method for unsteady navier – stokes solutions, Int. J. numer. meth. fluids, 33, 223-248, (2000) · Zbl 0976.76049
[22] Tuminaro, R.S.; Walker, H.F.; Shadid, J.N., On backtracking failure in Newton-GMRES methods with a demonstration for the navier – stokes equations, J. comput. phys., 180, 549-558, (2002) · Zbl 1143.76489
[23] 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
[24] Hunke, E.C., Viscous-plastic sea ice dynamics with the EVP model: linearization issues, J. comput. phys., 170, 18-38, (2001) · Zbl 1030.74032
[25] 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. amer. meteorol. soc., 77, 437-470, (1996)
[26] Akima, H., Rectangular-grid-data surface Fitting that has the accuracy of a bicubic polynomial, Trans. math. softw., 22, 3, 357-361, (1996) · Zbl 0884.65009
[27] Johnson, M.; Gaffigan, S.; Hunke, E.; Gerdes, R., A comparison of arctic Ocean sea ice concentration among the coordinated AOMIP model experiments, J. geophys. res., 112, C04S11, (2007)
[28] Maslowski, W.; Lipscomb, W.H., High resolution simulations of arctic sea ice, 1979-1993, Polar res., 22, 67-74, (2003)
[29] R. Kwok, Deformations of the Arctic Ocean sea ice cover between November 1996 and April 1997: a survey, J. Dempsey, H. Shen, L. Shapiro, 2001.
[30] Kelley, C.T., Iterative methods for linear and nonlinear equations, (1995), SIAM · Zbl 0832.65046
[31] Hibler, W.D.; Schulson, E.M., On modeling the anisotropic failure and flow of flawed sea ice, J. geophys. res., 105, C7, 17105-17120, (2000)
[32] Coffey, T.S.; Kelley, C.T.; Keyes, D.E., Pseudo-transient continuation and differential-algebraic equations, SIAM J. sci. comput., 25, 2, 553-569, (2003) · Zbl 1048.65080
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.