Pure and Applied Geophysics

, Volume 174, Issue 3, pp 1071–1089 | Cite as

On the Quality of Velocity Interpolation Schemes for Marker-in-Cell Method and Staggered Grids

  • Adina E. Pusok
  • Boris J. P. Kaus
  • Anton A. Popov
Article
  • 223 Downloads

Abstract

The marker-in-cell method is generally considered a flexible and robust method to model the advection of heterogenous non-diffusive properties (i.e., rock type or composition) in geodynamic problems. In this method, Lagrangian points carrying compositional information are advected with the ambient velocity field on an Eulerian grid. However, velocity interpolation from grid points to marker locations is often performed without considering the divergence of the velocity field at the interpolated locations (i.e., non-conservative). Such interpolation schemes can induce non-physical clustering of markers when strong velocity gradients are present (Journal of Computational Physics 166:218–252, 2001) and this may, eventually, result in empty grid cells, a serious numerical violation of the marker-in-cell method. To remedy this at low computational costs, Jenny et al. (Journal of Computational Physics 166:218–252, 2001) and Meyer and Jenny (Proceedings in Applied Mathematics and Mechanics 4:466–467, 2004) proposed a simple, conservative velocity interpolation scheme for 2-D staggered grid, while Wang et al. (Geochemistry, Geophysics, Geosystems 16(6):2015–2023, 2015) extended the formulation to 3-D finite element methods. Here, we adapt this formulation for 3-D staggered grids (correction interpolation) and we report on the quality of various velocity interpolation methods for 2-D and 3-D staggered grids. We test the interpolation schemes in combination with different advection schemes on incompressible Stokes problems with strong velocity gradients, which are discretized using a finite difference method. Our results suggest that a conservative formulation reduces the dispersion and clustering of markers, minimizing the need of unphysical marker control in geodynamic models.

Keywords

Interpolation Method Interpolation Scheme Stagger Grid Advection Scheme Subduction Channel 

Notes

Acknowledgements

We thank Taras Gerya for sharing the LinP interpolation method and for the many discussions related to this paper. We also thank the reviewers Thibault Duretz, Cedric Thieulot, Hongliang Wang, and the editor Wim Spakman, whose comments helped improve this paper. Funding was provided by the European Research Council under the European Community’s Seventh Framework Program (FP7/2007-2013)/ERC Grant Agreement #258830. The simulation data and the codes to reproduce the results presented in this study can be provided on request.

References

  1. Agrusta, R., van Hunen, J., & Goes, S. (2014). The effect of metastable pyroxene on the slab dynamics. Geophysical Research Letters, 41(24), 8800–8808.CrossRefGoogle Scholar
  2. Crameri, F., Schmeling, H., Golabek, G., Duretz, T., Orendt, R., Buiter, S., et al. (2012). A comparison of numerical surface topography calculations in geodynamic modelling: An evaluation of the ‘sticky air’ method. Geophysical Journal International, 189(1), 38–54.CrossRefGoogle Scholar
  3. Duretz, T., May, D., Gerya, T., & Tackley, P. (2011). Discretization errors and free surface stabilization in the finite difference and marker-in-cell method for applied geodynamics: A numerical study. Geochemistry Geophysics Geosystems, 12(7), 1–26.CrossRefGoogle Scholar
  4. Duretz, T., May, D., & Yamato, P. (2016). A free surface capturing discretization for the staggered grid finite difference scheme. Geophysical Journal International, 204(3), 1518–1530.CrossRefGoogle Scholar
  5. Evans, M., & Harlow, F. (1957). The particle-in-cell method for hydrodynamic calculations. Los Alamos National Laboratory Report, LA-2139.Google Scholar
  6. Fornberg, B. (1995). A practical guide to pseudospectral methods. In: Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press.Google Scholar
  7. Gerya, T. (2010). Introduction to numerical geodynamic modelling. Cambridge University Press.Google Scholar
  8. Gerya, T., & Yuen, D. (2003). Characteristics-based marker-in-cell method with conservative finite-differences schemes for modeling geological flows with strongly variable transport properties. Physics of the Earth and Planetary Interiors, 140, 293–318.CrossRefGoogle Scholar
  9. Harlow, F., & Welch, J. (1965). Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. The Physics of Fluids, 8(2), 2182–2189.CrossRefGoogle Scholar
  10. Hirth, G., & Kohlstedt, D. (2003). Rheology of the upper mantle and the mantle wedge: A view from the experimentalists (Vol. 138, pp. 83–105). Washington, D.C: American Geophysical Union.Google Scholar
  11. Ismail-Zadeh, A. and Tackley, P. J. (2010). Computational methods for geodynamics. Cambridge University Press.Google Scholar
  12. Jenny, P., Pope, S., Muradoglu, M., & Caughey, D. (2001). a hybrid algorithm for the joint pdf equation of turbulent reactive flows. Journal of Computational Physics, 166, 218–252.CrossRefGoogle Scholar
  13. Kaus, B., Mühlhaus, H., & May, D. (2010). A stabilization algorithm for geodynamic numerical simulations with a free surface. Physics of the Earth and Planetary Interiors, 181, 12–20.CrossRefGoogle Scholar
  14. Kaus, B., Popov, A., Baumann, T., Pusok, A., Bauville, A., Fernandez, N., & Collignon, M. (2016). Forward and inverse modeling of lithospheric deformation on geological timescales. NIC Symposium 2016---Proceedings, 48:1–8.Google Scholar
  15. McNamara, A., & Zhong, S. (2004). The influence of thermochemical convection on the fixity of mantle plumes. Earth and Planetary Science Letters, 222(2), 485–500.CrossRefGoogle Scholar
  16. Meyer, D., & Jenny, P. (2004). Conservative velocity interpolation for PDF methods. proceedings in applied mathematics and mechanics, 4, 466–467.CrossRefGoogle Scholar
  17. Mishin, Y. (2011). Adaptive multiresolution methods for problems of computational geodynamics. PhD thesis.Google Scholar
  18. Moresi, L., Dufour, F., & Mühlhaus, H. (2003). A Lagrangian integration point finite element method for large deformation modeling of viscoelastic geomaterials. Journal of Computational Physics, 184(2), 476–497.CrossRefGoogle Scholar
  19. Moresi, L., Zhong, S., & Gurnis, M. (1996). The accuracy of finite element solutions of Stokes’s flow with strongly varying viscosity. Physics of the Earth and Planetary Interiors, 97(1–4), 83–94.CrossRefGoogle Scholar
  20. Ranalli, G. (1995). Rheology of the earth (2nd ed.). London: Chapman and Hall.Google Scholar
  21. Schmeling, H., Babeyko, A., Enns, A., Faccenna, C., Funiciello, F., Gerya, T., et al. (2008). A benchmark comparison of spontaneous subduction models—towards a free surface. Physics of the Earth and Planetary Interiors, 171, 198–223.CrossRefGoogle Scholar
  22. Tackley, P., & King, S. (2003). Testing the tracer ratio method for modeling active compositional fields in mantle convection simulations. Geochemistry Geophysics Geosystems, 4(4), 1–15.CrossRefGoogle Scholar
  23. Thielmann, M., May, D., & Kaus, B. (2014). Discretization errors in the hybrid finite element particle-in-cell method. Pure and Applied Geophysics, 171(9), 2165–2184.CrossRefGoogle Scholar
  24. Turcotte, D. & Schubert, G. (2002). Geodynamics. Cambridge University Press, 2nd edn.Google Scholar
  25. van Keken, P., King, S., Schmeling, H., Christensen, U., Neumeister, D., & Doin, M. (1997). A comparison of methods for the modeling of thermochemical convection. Journal of Geophysical Research, 102(B10), 22477–22495.CrossRefGoogle Scholar
  26. Velić, M., May, D., & Moresi, L. (2008). A fast robust algorithm for computing discrete voronoi diagrams. Journal of Mathematical Modelling and Algorithms, 8(3), 343–355.Google Scholar
  27. Wang, H., Agrusta, R., & van Hunen, J. (2015). Advantages of a conservative velocity interpolation (CVI) scheme for particle-in-cell methods with application in geodynamic modeling. Geochemistry, Geophysics, Geosystems, 16(6), 2015–2023CrossRefGoogle Scholar
  28. Woidt, W. (1978). Finite-element calculations applied to salt dome analysis. Tectonophysics, 50, 369–386.CrossRefGoogle Scholar
  29. Zhong, S. (1996). Analytic solutions for Stokes’ flow with lateral variations in viscosity. Geophysical Journal International, 124, 18–28.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Institute of GeosciencesJohannes Gutenberg UniversityMainzGermany
  2. 2.Scripps Institution of OceanographyUC San DiegoSan DiegoUSA

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