Computational Methods in Science and Engineering

Achieving Textbook Multigrid Efficiency for Hydrostatic Ice Sheet Flow

Abstract

The hydrostatic equations for ice sheet flow offer improved fidelity compared with the shallow ice approximation and shallow stream approximation popular in today's ice sheet models. Nevertheless, they present a serious bottleneck because they require the solution of a three-dimensional (3D) nonlinear system, as opposed to the two-dimensional system present in the shallow stream approximation. This 3D system is posed on high-aspect domains with strong anisotropy and variation in coefficients, making it expensive to solve with current methods. This paper presents a Newton--Krylov multigrid solver for the hydrostatic equations that demonstrates textbook multigrid efficiency (an order of magnitude reduction in residual per iteration and solution of the fine-level system at a small multiple of the cost of a residual evaluation). Scalability on Blue Gene/P is demonstrated, and the method is compared to various algebraic methods that are in use or have been proposed as viable approaches.

Keywords

  1. hydrostatic
  2. ice sheet
  3. Newton--Krylov
  4. multigrid
  5. preconditioning

MSC codes

  1. 86-08
  2. 65N22
  3. 65Y05

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References

1.
S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, PETSc Users Manual, Technical report ANL-95/11 - Revision 3.3, Argonne National Laboratory, Lemont, IL, 2012.
2.
S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, PETSc Web page, http://www.mcs.anl.gov/petsc, 2012.
3.
L. Bernstein et al., Climate Change 2007: Synthesis Report, Intergovernmental Panel on Climate Change, 2008, www.ipcc.ch/ipccreports/ar4-syr.htm.
4.
H. Blatter, Velocity and stress fields in grounded glaciers: A simple algorithm for including deviatoric stress gradients, J. Glaciol., 41 (1995), pp. 333--344.
5.
E. Bueler and J. Brown, Shallow shelf approximation as a “sliding law” in a thermomechanically coupled ice sheet model, J. Geophys. Res., 114 (2009), F03008.
6.
X. Cai and M. Sarkis, A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput., 21 (1999), pp. 792--797.
7.
S. Chow, G. Carey, and M. Anderson, Finite element approximations of a glaciology problem, M2AN Math. Model. Numer. Anal., 38 (2004), pp. 741--756.
8.
J. Colinge and J. Rappaz, A strongly nonlinear problem arising in glaciology, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 395--406.
9.
B. De Smedt, F. Pattyn, and P. De Groen, Using the unstable manifold correction in a Picard iteration to solve the velocity field in higher-order ice-flow models, J. Glaciol., 56 (2010), pp. 257--261.
10.
S. C. Eisenstat and H. F. Walker, Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17 (1996), pp. 16--32.
11.
K. Evans, A. Salinger, E. Barker, D. Holland, D. Knoll, J. F. LeMieux, B. Lipscomb, R. Nong, S. Price, K. Roddy, T. White, and P. Worley, A Scalable, Efficient, and Accurate Community Ice Sheet Model, http://www.csm.ornl.gov/SEACISM, 2010.
12.
A. C. Fowler, Modelling the flow of glaciers and ice sheets, in Continuum Mechanics and Applications in Geophysics and the Environment, Springer, Berlin, 2001, pp. 201--221.
13.
M. Gee, C. Siefert, J. Hu, R. Tuminaro, and M. Sala, ML $5.0$ Smoothed Aggregation User's Guide, Technical report SAND2006-2649, Sandia National Laboratories, Albuquerque, NM, 2006.
14.
R. Glowinski and J. Rappaz, Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology, M2AN Math. Model. Numer. Anal., 37 (2003), pp. 175--186.
15.
D. Goldberg, A variationally derived, depth-integrated approximation to a higher-order glaciological flow model, J. Glaciol., 57 (2011), pp. 157--170.
16.
D. Goldberg, D. Holland, and C. Schoof, Grounding line movement and ice shelf buttressing in marine ice sheets, J. Geophys. Res., 114 (2009), F04026.
17.
D. Goldsby and D. Kohlstedt, Superplastic deformation of ice: Experimental observations, J. Geophys. Res., 106 (2001), pp. 11017--11030.
18.
R. Greve and H. Blatter, Dynamics of Ice Sheets and Glaciers, Springer-Verlag, Berlin, 2009.
19.
V. Henson and U. Yang, BoomerAMG: A parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41 (2002), pp. 155--177.
20.
R. Hindmarsh, A numerical comparison of approximations to the Stokes equations used in ice sheet and glacier modeling, J. Geophys. Res., 109 (2004), F01012.
21.
K. Hutter, Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets, Springer, 1983.
22.
J. Johnson and J. Staiger, Modeling long-term stability of the Ferrar Glacier, East Antarctica: Implications for interpreting cosmogenic nuclide inheritance, J. Geophys. Res., 112 (2007), F03S30.
23.
E. Larour, M. Morlighem, and H. Seroussi, Ice Sheet System Model, 2010, http://issm.jpl. nasa.gov.
24.
E. Larour, H. Seroussi, M. Morlighem, and E. Rignot, Modeling the flow of glaciers in steep terrains: The integrated second-order shallow ice approximation (iSOSIA), J. Geophys. Res., 117 (2012), F02009.
25.
J.-F. Lemieux, S. F. Price, K. J. Evans, D. Knoll, A. G. Salinger, D. M. Holland, and A. J. Payne, Implementation of the Jacobian-free Newton-Krylov method for solving the first-order ice sheet momentum balance, J. Comput. Phys., 230 (2011), pp. 6531--6545.
26.
L. Morland, Unconfined ice-shelf flow, in Dynamics of the West Antarctic Ice Sheet, C. van der Veen and J. Oerlemans, eds., Kluwer Academic Publishers, Norwell, MA, 1987, pp. 99--116.
27.
W. Paterson, Physics of Glaciers, 3rd ed., Oxford: Butterworth-Heinemann, Woburn, MA, 1998.
28.
F. Pattyn, Transient glacier response with a higher-order numerical ice-flow model, J. Glaciol., 48 (2002), pp. 467--477.
29.
F. Pattyn, A new three-dimensional higher-order thermomechanical ice sheet model: Basic sensitivity, ice stream development, and ice flow across subglacial lakes, J. Geophys. Res, 108 (2003), 2382.
30.
F. Pattyn, L. Perichon, A. Aschwanden, B. Breuer, B. De Smedt, O. Gagliardini, G. Gudmundsson, R. Hindmarsh, A. Hubbard, J. Johnson, T. Kleiner, Y. Konovalov, C. Martin, A. Payne, D. Pollard, S. Price, M. Rückamp, F. Saito, O. Souček, S. Sugiyama, and T. Zwinger, Benchmark experiments for higher-order and full Stokes ice sheet models (ISMIP-HOM), The Cryosphere, 2 (2008), pp. 95--108.
31.
C. Raymond, Energy balance of ice streams, J. Glaciol., 46 (2000), pp. 665--674.
32.
C. Schoof, A variational approach to ice stream flow, J. Fluid Mech., 556 (2006), pp. 227--251.
33.
C. Schoof and R. Hindmarsh, Thin-film flows with wall slip: an asymptotic analysis of higher order glacier flow models, Quart. J. Mech. Appl. Math., 63 (2010), pp. 73--114.
34.
B. Smith, P. Bjørstad, and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1996.
35.
M. Truffer and K. Echelmeyer, Of Isbr\ae and ice streams, Ann. Glaciol., 36 (2003), pp. 66--72.
36.
H. Tufo and P. Fischer, Fast parallel direct solvers for coarse grid problems, J. Parallel Distrib. Comput., 61 (2001), pp. 151--177.
37.
J. Weertman, On the sliding of glaciers, J. Glaciol., 3 (1957), pp. 33--38.
38.
M. Weis, R. Greve, and K. Hutter, Theory of shallow ice shelves, Continuum Mech. Thermodyn., 11 (1999), pp. 15--50.

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Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: B359 - B375
ISSN (online): 1095-7197

History

Submitted: 18 May 2011
Accepted: 2 July 2012
Published online: 12 March 2013

Keywords

  1. hydrostatic
  2. ice sheet
  3. Newton--Krylov
  4. multigrid
  5. preconditioning

MSC codes

  1. 86-08
  2. 65N22
  3. 65Y05

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