A parallel Jacobian-free Newton-Krylov solver for a coupled sea ice-ocean model. (English) Zbl 1349.86008

Summary: The most common representation of sea ice dynamics in climate models assumes that sea ice is a quasi-continuous non-normal fluid with a viscous-plastic rheology. This rheology leads to non-linear sea ice momentum equations that are notoriously difficult to solve. Recently a Jacobian-free Newton-Krylov (JFNK) solver was shown to solve the equations accurately at moderate costs. This solver is extended for massive parallel architectures and vector computers and implemented in a coupled sea ice-ocean general circulation model for climate studies. Numerical performance is discussed along with numerical difficulties in realistic applications with up to 1920 CPUs. The parallel JFNK-solver’s scalability competes with traditional solvers although the collective communication overhead starts to show a little earlier. When accuracy of the solution is required (i. e. reduction of the residual norm of the momentum equations of more that one or two orders of magnitude) the JFNK-solver is unrivalled in efficiency. The new implementation opens up the opportunity to explore physical mechanisms in the context of large scale sea ice models and climate models and to clearly differentiate these physical effects from numerical artifacts.


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
76T99 Multiphase and multicomponent flows
86A05 Hydrology, hydrography, oceanography
86A60 Geological problems


Full Text: DOI


[1] Proshutinsky, A.; Kowalik, Z., Preface to special section on arctic Ocean model intercomparison project (AOMIP) studies and results, J. Geophys. Res., 112, (2007)
[2] Rampal, P.; Weiss, J.; Dubois, C.; Campin, J.-M., IPCC climate models do not capture arctic sea ice drift acceleration: consequences in terms of projected sea ice thinning and decline, J. Geophys. Res., 116, (2011)
[3] Hibler, W. D., A dynamic thermodynamic sea ice model, J. Phys. Oceanogr., 9, 815-846, (1979)
[4] Wilchinsky, A. V.; Feltham, D. L., Modelling the rheology of sea ice as a collection of diamond-shaped floes, J. Non-Newton. Fluid Mech., 138, 22-32, (2006) · Zbl 1195.74112
[5] Feltham, D. L., Sea ice rheology, Annu. Rev. Fluid Mech., 40, 91-112, (2008) · Zbl 1136.76007
[6] Tsamados, M.; Feltham, D. L.; Wilchinsky, A. V., Impact of a new anisotropic rheology on simulations of arctic sea ice, J. Geophys. Res., 118, 91-107, (2013)
[7] Wang, K.; Wang, C., Modeling linear kinematic features in pack ice, J. Geophys. Res., 114, (2009)
[8] Girard, L.; Bouillon, S.; Jérôme, W.; Amitrano, D.; Fichefet, T.; Legat, V., A new modelling framework for sea ice models based on elasto-brittle rheology, Ann. Glaciol., 52, 123-132, (2011)
[9] Lemieux, J.-F.; Tremblay, B., Numerical convergence of viscous-plastic sea ice models, J. Geophys. Res., 114, (2009)
[10] Lemieux, J.-F.; Knoll, D.; 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)
[11] Zhang, J.; Hibler, W. D., On an efficient numerical method for modeling sea ice dynamics, J. Geophys. Res., 102, 8691-8702, (1997)
[12] 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)
[13] 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)
[14] 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
[15] Cai, X.-C.; Sarkis, M., A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput., 21, 792-797, (1999) · Zbl 0944.65031
[16] Hovland, P. D.; McInnes, L. C., Parallel simulation of compressible flow using automatic differentiation and petsc, Parallel Comput., 27, 503-519, (2001) · Zbl 0972.68165
[17] 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
[18] Paniconi, C.; Putti, M., A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems, Water Resour. Res., 30, 3357-3374, (1994)
[19] Marshall, J.; Adcroft, A.; Hill, C.; Perelman, L.; Heisey, C., A finite-volume, incompressible Navier Stokes model for studies of the Ocean on parallel computers, J. Geophys. Res., 102, 5753-5766, (1997) · Zbl 0907.58089
[20] Mitgcm user manual, (2012), Online documentation, MIT/EAPS, Cambridge, MA 02139, USA
[21] Hundsdorfer, W.; Trompert, R. A., Method of lines and direct discretization: a comparison for linear advection, Appl. Numer. Math., 13, 469-490, (1994) · Zbl 0797.65067
[22] Roe, P., Some contributions to the modelling of discontinuous flows, (Engquist, B.; Osher, S.; Somerville, R., Large-Scale Computations in Fluid Mechanics, Lectures in Applied Mathematics, vol. 22, (1985), American Mathematical Society), 163-193
[23] Geiger, C. A.; Hibler, W. D.; Ackley, S. F., Large-scale sea ice drift and deformation: comparison between models and observations in the western weddell sea during 1992, J. Geophys. Res., 103, 21893-21913, (1998)
[24] 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, (2008)
[25] Hunke, E. C.; Dukowicz, J. K., The elastic-viscous-plastic sea ice dynamics model in general orthogonal curvilinear coordinates on a sphere—incorporation of metric terms, Mon. Weather Rev., 130, 1847-1865, (2002)
[26] Knoll, D.; Keyes, D., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 357-397, (2004) · Zbl 1036.65045
[27] Giering, R.; Kaminski, T., Recipes for adjoint code construction, ACM Trans. Math. Softw., 24, 437-474, (1998) · Zbl 0934.65027
[28] Heimbach, P.; Hill, C.; Giering, R., An efficient exact adjoint of the parallel MIT general circulation model, generated via automatic differentiation, Future Gener. Comput. Syst., 21, 1356-1371, (2005)
[29] Saad, Y., A flexible inner-outer preconditioned GMRES method, SIAM J. Sci. Comput., 14, 461-469, (1993) · Zbl 0780.65022
[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] Cai, X.-C.; Gropp, W. D.; Keyes, D. E.; Tidriri, M. D., Newton-Krylov-Schwarz methods in CFD, (Hebeker, F.-K.; Rannacher, R.; Wittum, G., Proceedings of the International Workshop on the Navier-Stokes Equations, (2013)), 17-30 · Zbl 0876.76059
[32] Cai, X.-C.; Gropp, W. D.; Keyes, D. E.; Melvin, R. G.; Young, D. P., Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation, SIAM J. Sci. Comput., 19, 246-265, (1998) · Zbl 0917.76035
[33] Thomas, L. H., Elliptic problems in linear differential equations over a network, (1949), Columbia University New York, Watson Sci. Comput. Lab Report
[34] Duff, I. S.; Meurant, G. A., The effect of ordering on preconditioned conjugate gradients, BIT Numer. Math., 29, 635-657, (1989) · Zbl 0687.65037
[35] Nguyen, A. T.; Kwok, R.; Menemenlis, D., Source and pathway of the western arctic upper halocline in a data-constrained coupled Ocean and sea ice model, J. Phys. Oceanogr., 43, 80-823, (2012)
[36] Rignot, E.; Fenty, I.; Menemenlis, D.; Xu, Y., Spreading of warm Ocean waters around greenland as a possible cause for glacier acceleration, Ann. Glaciol., 53, 257-266, (2012)
[37] Karcher, M.; Beszczynska-Möller, A.; Kauker, F.; Gerdes, R.; Heyen, S.; Rudels, B.; Schauer, U., Arctic Ocean warming and its consequences for the Denmark strait overflow, J. Geophys. Res., 116, (2011)
[38] Hill, C.; Menemenlis, D.; Ciotti, B.; Henze, C., Investigating solution convergence in a global Ocean model using a 2048-processor cluster of distributed shared memory machines, Sci. Program., 12, 107-115, (2007)
[39] Schlichting, J. J.F. M.; van der Vorst, H. A., Solving 3D block bidiagonal linear systems on vector computers, J. Comput. Appl. Math., 27, 323-330, (1989) · Zbl 0679.65021
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.