Skip to main content
SpringerLink
Log in

Multilevel and local time-stepping discontinuous Galerkin methods for magma dynamics

  • ORIGINAL PAPER
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

Discontinuous Galerkin (DG) method is presented for numerical modeling of melt migration in a chemically reactive and viscously deforming upwelling mantle column at local chemical equilibrium. DG methods for both advection and elliptic equations provide a robust and efficient solution to the problems of melt migration in the asthenospheric upper mantle. Assembling and solving the elliptic equation is the major bottleneck in these computations. To address this issue, adaptive mesh refinement and local time-stepping methods have been proposed to improve the computational wall time. The robustness of DG methods is demonstrated through two benchmark problems by modeling detailed structure of high-porosity dissolution channels and compaction dissolution waves.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aharonov, E., Whitehead, J., Kelemen, P.B., Spiegelman, M.: Channeling instability of upwelling melt in the mantle. J. Geophys. Res. 100, 433–450 (1995)

    Google Scholar 

  2. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)

    Article  Google Scholar 

  3. Bangerth, W., Hartmann, R., Kanschat, G.: deal.II: a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33(4), 24/1–24/27 (2007)

    Article  Google Scholar 

  4. Bercovici, D., Ricard, Y., Schubert, G.: A two-phase model for compaction and damage 1. General theory. J. Geophys. Res. 106(B5), 8887–8906 (2001)

    Article  Google Scholar 

  5. Butler, S.L.: The effects of buoyancy on shear-induced melt bands in a compacting porous medium. Phys. Earth Planet. Inter. 173, 51–59 (2009)

    Article  Google Scholar 

  6. Castro, C.E., Kaser, M., Toro, E.F.: Space-time adaptive numerical methods for geophysical applications. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 367, 4613–4631 (2009)

    Article  Google Scholar 

  7. Chueh, C.C., Secanell, M., Bangerth, W., Djilali, N.: Multi-level adaptive simulation of transient two-phase flow in heterogeneous porous media. Comput. Fluids 39, 1585–1596 (2010)

    Article  Google Scholar 

  8. Cockburn, B., Shu, C.W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)

    Article  Google Scholar 

  9. Cockburn, B., Shu, C.W.: The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws. Math. Model Numer. Anal. 25, 337–361 (1991)

    Google Scholar 

  10. Cockburn, B., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)

    Google Scholar 

  11. Cockburn, B., Lin, S.Y., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84, 90–113 (1989)

    Article  Google Scholar 

  12. Cockburn, B., Shu, C.W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545–581 (2001)

    Google Scholar 

  13. Dumbser, M., Kaser, M., Toro, E.F.: An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes-V. Local time stepping and p-adaptivity. Geophys. J. Int. 171(2), 695–717 (2007)

    Article  Google Scholar 

  14. Gear, C., Wells, D.: Multirate linear multistep methods. BIT 24, 484–502 (1984)

    Article  Google Scholar 

  15. Ghods, A., Arkani-Hamed, J.: Melt migration beneath mid-ocean ridges. Geophys. J. Int. 140, 687–697 (2000)

    Article  Google Scholar 

  16. Giraldo, F.X., Hesthaven, J.S., Warburton, T.: Nodal high-order discontinuous Galerkin method for the spherical shallow water equations. J. Comput. Phys. 181, 499–525 (2002)

    Article  Google Scholar 

  17. Goedel, N., Schomann, S., Warburton, T., Clemens, M.: Local timestepping discontinuous Galerkin methods for electromagnetic RF field Problems. EuCAP (2009)

  18. Hesse, M.A., Schiemenz, A.R., Liang, Y., Parmentier, E.M.: Compaction dissolution waves in an upwelling mantle column. Geophys. J. Int. 187, 1057–1075 (2011)

    Article  Google Scholar 

  19. Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral methods for time dependent problems, vol. 21. Cambridge University Press (2007)

  20. Hesthaven, J.S., Warburton, T.: Nodal discontinuous Galerkin methods: Algorithms, analysis, and applications, vol. 54. Springer (2008)

  21. Hesthaven, J.S., Warburton, T., Chauviere, C., Wilcox, L.: High-order discontinuous Galerkin methods for computational electromagnetics and uncertainty quantification. Sci. Comput. Electr. Eng., 403–412 (2010)

  22. Katz, R.: Porosity-driven convection and asymmetry beneath mid-ocean ridges. Geochem. Geophys. Geosyst. 11(11) (2010)

  23. Keller, T., May, D.A., Kaus, B.J.P.: Numerical modelling of magma dynamics coupled to tectonic deformation of lithosphere and crust. Geophys. J. Int. 186, 641–664 (2013)

    Google Scholar 

  24. Liang, Y., Schiemenz, A., Hesse, M.A., Parmentier, E.M., Hesthaven, J.S.: High-porosity channels for melt migration in the mantle: top is the dunite and bottom is the harzburgite and lherzolite. Geophys. Res. Lett. 37(15) (2010)

  25. Liang, Y., Schiemenz, A., Hesse, M.: Waves, channels, and the preservation of chemical heterogeneities during melt migration in the mantle. Geophys. Res. Lett. 38(20) (2011)

  26. McKenzie, D.: The generation and compaction of partially molten rocks. J. Petrol. 25, 713–765 (1984)

    Article  Google Scholar 

  27. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Los Alamos Report LA-UR-73-479 (1973)

  28. Richter, A., Stiller, J., Grundmann, R.: Stabilized high-order discontinuous Galerkin methods for aeroacoustic Investigations. Comput. Fluid Dyn., 77–82 (2009)

  29. Riviere, B.: Discontinuous Galerkin methods For solving elliptic And parabolic equations. Frontiers in Applied Mathematics (2008)

  30. Saad, Y.: Iterative methods for sparse linear systems (2nd edition). SIAM Publishing (2003)

  31. Schiemenz, A.R., Hesse, M.A., Hesthaven, J.S.: Modeling magma dynamics with a mixed Fourier collocation-discontinuous Galerkin method. Commun. Comput. Phys. 10(2), 433–452 (2009)

    Google Scholar 

  32. Schiemenz, A.R., Liang, Y., Parmentier, E.M.: A high-order numerical study of reactive dissolution in an upwelling heterogeneous mantle: I. Channelization, channel lithology, and channel geometry. Geophys. J. Int. 186, 641–664 (2011)

    Article  Google Scholar 

  33. Scott, D.R., Stevenson, D.J.: Magma ascent by porous flow. J. Geophys. Res. 91, 9283–9296 (1986)

    Article  Google Scholar 

  34. Spiegelman, M., Katz, R.: A semi-Lagrangian Crank-Nicolson algorithm for the numerical solution of advection-diffusion problems. Geochem. Geophys. Geosyst. 7, Q04014 (2006)

    Article  Google Scholar 

  35. Spiegelman, M., Kelemen, P.B., Aharonov, E.: Causes and consequences of flow organization during melt transport: The reaction infiltration instability in compactible media. J. Geophys. Res. 106, 2061–2077 (2001)

    Article  Google Scholar 

  36. Steefel, C., Lasaga, A.: Evolution of dissolution patterns, in Chemical Modeling in Aqueous Systems II. Am. Chem. Soc. 106, 212–225 (1990)

    Google Scholar 

  37. Sun, S., Wheeler, M.F.: Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal. 43(1), 195–219 (2005)

    Article  Google Scholar 

  38. Tirupathi, S.: Discontinuous Galerkin methods for magma dynamics. Brown Digital Repository

  39. Tirupathi, S., Hesthaven, J.S., Liang, Y.: Modeling 3D magma dynamics using a discontinuous Galerkin method. Commun. Comput. Phys. (2015). (to appear)

  40. Van der Vegt, J., Van der Ven, J.J.W.: Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows: I. General formulation. J. Comput. Phys. 182(2), 546–585 (2002)

    Article  Google Scholar 

  41. Zingan, V., Guermond, J.L., Morel, J., Popov, B.: Implementation of the entropy viscosity method with the discontinuous Galerkin method. Comput. Methods Appl. Mech. Eng. 253, 479–490 (2012)

    Article  Google Scholar 

Download references

Author information

Author notes

  1. S. Tirupathi

    Present address: IBM Research - Ireland, Damastown Industrial Park, Dublin, Ireland

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, USA

    S. Tirupathi

  2. Mathematics Institute of Computational Science and Engineering, EPFL, Lausanne, Switzerland

    J. S. Hesthaven

  3. Department of Earth, Environmental and Planetary Sciences, Brown University, Providence, USA

    Y. Liang & M. Parmentier

Authors
  1. S. Tirupathi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. S. Hesthaven
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Y. Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Parmentier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Tirupathi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tirupathi, S., Hesthaven, J.S., Liang, Y. et al. Multilevel and local time-stepping discontinuous Galerkin methods for magma dynamics. Comput Geosci 19, 965–978 (2015). https://doi.org/10.1007/s10596-015-9514-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-015-9514-7

Keywords

Search

Navigation