Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T03:03:20.229Z Has data issue: false hasContentIssue false
  • English
  • Français

Coupled Models and Parallel Simulations for Three-Dimensional Full-Stokes Ice Sheet Modeling

Published online by Cambridge University Press:  28 May 2015

Huai Zhang ,
Lili Ju ,
Max Gunzburger ,
Todd Ringler  and
Stephen Price
Huai Zhang*
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA Laboratory of Computational Geodynamics, Graduate University of Chinese Academy of Sciences, Beijing, 100049, China
Lili Ju*
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
Max Gunzburger*
Affiliation:
Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, USA
Todd Ringler*
Affiliation:
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Stephen Price*
Affiliation:
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
*
Corresponding author.Email address:hzhang@gucas.ac.cn
Corresponding author.Email address:ju@math.sc.edu
Corresponding author.Email address:mgunzburger@fsu.edu
Corresponding author.Email address:ringler@lanl.gov
Corresponding author.Email address:sprice@lanl.gov
Get access

Abstract

A three-dimensional full-Stokes computational model is considered for determining the dynamics, temperature, and thickness of ice sheets. The governing thermo-mechanical equations consist of the three-dimensional full-Stokes system with nonlinear rheology for the momentum, an advective-diffusion energy equation for temperature evolution, and a mass conservation equation for ice-thickness changes. Here, we discuss the variable resolution meshes, the finite element discretizations, and the parallel algorithms employed by the model components. The solvers are integrated through a well-designed coupler for the exchange of parametric data between components. The discretization utilizes high-quality, variable-resolution centroidal Voronoi Delaunay triangulation meshing and existing parallel solvers. We demonstrate the gridding technology, discretization schemes, and the efficiency and scalability of the parallel solvers through computational experiments using both simplified geometries arising from benchmark test problems and a realistic Greenland ice sheet geometry.

Keywords

65M6065M5565Y0565Z0568U2068W10Ice sheet modelingnonlinear Stokes equationfinite element methodparallel implementationcentroial Voronoi Delaunay meshes
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 3 , August 2011 , pp. 396 - 418
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Agullo, E., Guermouche, A., and L’excellent, J., A preliminary out-of-core extension of a parallel multifrontal solver, Euro-Par 2006 Parallel Processing, pp. 1053–1063.CrossRefGoogle Scholar
[2]Ainsworth, M., A preconditioner based on domain decomposition for h-p finite-element approximation on quasi-uniform meshes, SIAM J. Numer. Anal. 33 1996, pp. 1358–1376.CrossRefGoogle Scholar
[3]Amestoy, P., Duff, I., L’excellent, J., and Koster, J., A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl. 23 2001, pp. 15–41.CrossRefGoogle Scholar
[4]Axelsson, O. and Larin, M., An algebraic multilevel iteration method for finite element matrices, J. Comput. Appl. Math. 89 1998, pp. 135–153.CrossRefGoogle Scholar
[5]Bamber, J., Ekholm, S., and Krabill, W., A new, high resolution digital elevation model of Greenland fully validated with airborne laser altimeter data, J. Geophys. Res. 106 2001, pp. 6733–6745.CrossRefGoogle Scholar
[6]Bengtsson, L., Climate of the 21st-Century, Agric. Forest Meteor. 72 1994, pp. 3–29.CrossRefGoogle Scholar
[7]Blatter, H., Velocity and stress fields in grounded glaciers: a simple algorithm for including deviatoric stress gradients, J. Glaciol. 41 1995, pp. 33–44.CrossRefGoogle Scholar
[8]Bromwich, D., Bai, L., and Bjarnason, G., High-resolution regional climate simulations over Iceland using Polar MM5, Month. Weath. Rev. 133 2005, pp. 3527–3547.CrossRefGoogle Scholar
[9]Douglas, C., Multigrid methods in science and engineering, IEEE Comput. Sci. Engrg. 3 1996, pp. 55–68.CrossRefGoogle Scholar
[10]Du, Q., Faber, V., and Gunzburger, M., Centroidal Voronoi tessellations: Applications and algorithms, SIAM Review 41 1999, pp. 637–676.CrossRefGoogle Scholar
[11]Du, Q., Gunzburger, M., and Ju, L., Advances in studies and applications of centroidal Voronoi tessellations, Numer. Math. Theor. Meth. Appl. 3 2010, pp. 119–142.CrossRefGoogle Scholar
[12]Du, Q. and Ju, L., Finite volume methods on spheres and spherical centroidal Voronoi meshes, SIAMJ. Numer. Anal. 43 2005, pp. 1673–1692.CrossRefGoogle Scholar
[13]Gregory, J. and Huybrechts, P., Ice-sheet contributions to future sea-level change, Philo. Trans. Royal Soc. - Math., Phys. Engrg. Sci. 2006 364, pp. 1709–1731.Google ScholarPubMed
[14]Gunzburger, M., Finite Element Methods for the Navier-Stokes Equations, Academic, Boston, 1989.Google Scholar
[15]Hanna, E., Huybrechts, P., Janssens, I., Cappelen, J., Steffen, K., and Stephens, A., Runoff and mass balance of the Greenland ice sheet: 1958-2003, J. Geophys. Res. 110 2005, D13108, doi:10.1029/2004JD005641.Google Scholar
[16]Hindmarsh, R. and Payne, A., Time-step limits for stable solutions of the ice-sheet equation, Ann. Glaciol. 23 1996, pp. 74–85.CrossRefGoogle Scholar
[17]Nguyen, H., Burkardt, J., Gunzburger, M., Ju, L., and Saka, Y., Constrained CVT meshes and a comparison of triangular mesh generators, Comp. Geom. Theo. Appl. 42 2009, pp. 1–19.CrossRefGoogle Scholar
[18]Holland, D., Ingram, R., Mysak, L., and Oberhuber, J., A numerical simulation of the sea-ice cover in the Northern Greenland sea, J. Geophys. Res. - Oceans 100 1995, pp. 4751–4760.CrossRefGoogle Scholar
[19]Holland, D., Mysak, L., Manak, D., and Oberhuber, J., Sensitivity study of a dynamic thermo-dynamic sea ice model, J. Geophys. Res. - Oceans 98 1993, pp. 2561–2586.CrossRefGoogle Scholar
[20]Ito, K. and Toivanen, J., Preconditioned iterative methods on sparse subspaces, Appl. Math. Lett. 19 2006, pp. 1191–1197.CrossRefGoogle Scholar
[21]Ju, L., Conforming centroidal Voronoi Delaunay triangulation for quality mesh generation, Inter. J. Numer. Anal. Model. 4 2007, pp. 531–547.Google Scholar
[22]Ju, L., Gunzburger, M., and Zhao, W.-D., Adaptive finite element methods for elliptic PDEs based on conforming centroidal Voronoi Delaunay triangulations, SIAM J. Sci. Comput. 28 2006, pp. 2023–2053.CrossRefGoogle Scholar
[23]Ju, L., Lee, H.-C., and Tian, L., Numerical simulations of the steady Navier-Stokes equations using adaptive meshing schemes, Inter. J. Numer. Meth. Fluids 56 2008, pp. 703–721.CrossRefGoogle Scholar
[24]Ju, L., Ringler, T., and Gunzburger, M., Voronoi tessellations and their application to climate and global modeling, Numerical Techniques for Global Atmospheric Models, Springer, 2011, pp. 313–342.CrossRefGoogle Scholar
[25]Jung, M., On the parallelization of multi-grid methods using a non-overlapping domain decomposition data structure, Appl. Numer. Math. 23 1997, pp. 119–137.CrossRefGoogle Scholar
[26]Karypis, G., Multi-constraint mesh partitioning for contact/impact computations, Supercom- puting’03, Phoenix, Arizona, USA.Google Scholar
[27]Korneev, V. and Jensen, S., Domain decomposition preconditioning in the hierarchical p-version of the finite element method, Appl. Numer. Math. 29 1999, pp. 479–518.CrossRefGoogle Scholar
[28]Le meur, E., Gagliardini, O., Zwinger, T., and Ruokolainen, J., Glacier flow modelling: A comparison of the shallow ice approximation and the full-Stokes solution, Compt. Rendus Phys. 5 2004, pp. 709–722.CrossRefGoogle Scholar
[29]Li, X. and Demmel, J., SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems, ACM Trans. Math. Soft. 29 2003, pp. 110–140.CrossRefGoogle Scholar
[30]Li, Z.-Z., Saad, Y., and Sosonkina, M., pARMS: A parallel version of the algebraic recursive multilevel solver, Numer. Lin. Alg Appl. 10 2003, pp. 485–509.CrossRefGoogle Scholar
[31]Lipscomb, W., Bindschadler, R., Bueler, E., Holland, D., Johnson, J., and Price, S., A community ice sheet model for sea level prediction, Eos 90 2009, pp. 23.CrossRefGoogle Scholar
[32]Mandel, J. and Tezaur, R., Convergence of a substructuring method with Lagrange multipliers, Numer. Math. 73 1996, pp. 473–487.CrossRefGoogle Scholar
[33]May, D. and Moresi, L., Preconditioned iterative methods for Stokes flow problems arising in computational geodynamics, Phys. Earth Planet. Inter. 171 2008, pp. 33–47, doi:10.1016/j.pepi.2008.07.036.CrossRefGoogle Scholar
[34]Mernild, S., Liston, G., Hiemstra, C., and Steffen, K., Surface melt area and water balance modeling on the Greenland ice sheet 1995-2005, J. Hydrometeor. 9 2008, pp. 1191–1211.CrossRefGoogle Scholar
[35]Mernild, S., Liston, G., Kane, D., Knudsen, N., and Hasholt, B., Snow, runoff, and mass balance modeling for the entire Mittivakkat Glacier (1998-2006), Ammassalik Island, SE Greenland, Geografisk Tidsskrift-Danish J. Geogra. 108 2008, pp. 121–136.CrossRefGoogle Scholar
[36]Michel, J., Pellegrini, F., and Roman, J., Unstructured graph partitioning for sparse linear system solving, Solving Irregularly Structured Problems in Parallel, Springer, Berlin, pp. 273–286.Google Scholar
[37]Nick, F., Vieli, A., Howat, I., and Joughin, I., Large-scale changes in Greenland outlet glacier dynamics triggered at the terminus, Nat Geosci. 2 2009, pp. 110–114.CrossRefGoogle Scholar
[38]Nye, J., The distribution of stress and velocity in glaciers and ice sheets, Proc. Roy. Soc. London - Ser. A 239 1957, pp. 113–133.Google Scholar
[39]Paterson, W., The Physics of Glaciers, oElsevier Science, Oxford, 1994.Google Scholar
[40]Pattyn, F., A new 3D higher-order thermomechanical ice-sheet model: Basic sensitivity, ice-stream development and ice flow across subglacial lakes, J. Geophys. Res. 108 2003, 2382, doi:10.1029/2002JB002329.Google Scholar
[41]Pattyn, F., Perichon, L., Aschwanden, A., Breuer, B., Smedt, B. De, Gagliardini, O., Gud-Mundsson, G., Hindmarsh, R., Hubbard, A., Johnson, J., Kleiner, T., Konovalov, Y., Martin, C., Payne, A., Pollard, D., Price, S., Ruckamp, M., Saito, F., Sugiyama, S., and Zwinger, T., Benchmark experiments for higher-order and full-Stokes ice sheet models (ISMIP-HOM), The Cryosphere 2 2008, pp. 95–108.CrossRefGoogle Scholar
[42]Price, S., Waddington, E., and Conway, H., A full-stress, thermomechanical flow band model using the finite volume method, J. Geophy Res. - Earth Surface 112 2007, F03020, doi:10.1029/2006JF000724.CrossRefGoogle Scholar
[43]Ringler, T., Ju, L., and Gunzburger, M., A multiresolution method for climate system modeling: Application of spherical centroidal Voronoi tessellations, Ocean Dyn. 58 2008, pp. 475–498.CrossRefGoogle Scholar
[44]Shadid, J. and Tuminaro, R., Sparse iterative algorithm software for large-scale MIMD machines - An initial discussion and implementation. Concur. Comput: Pract Exper. 4 1992, pp. 481–497.Google Scholar
[45]Valli, A., Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications, Oxford, 1999.Google Scholar
[46]Zumbusch, G., Schur Complement Domain Decomposition Methods in Diffpack, Technical report, Sintef Applied Mathematics, Oslo, Norway, 1996.Google Scholar