Articles | Volume 17, issue 10
https://doi.org/10.5194/gmd-17-4115-2024
© Author(s) 2024. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/gmd-17-4115-2024
© Author(s) 2024. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Development and technical paper
| 
21 May 2024
Development and technical paper |
| 21 May 2024
| 21 May 2024
Benchmarking the accuracy of higher-order particle methods in geodynamic models of transient flow
Department of Geological Sciences, University of Florida, Gainesville, Florida, USA
now at: GEOMAR Helmholtz Centre for Ocean Research Kiel, Kiel, Germany
Juliane Dannberg
Department of Geological Sciences, University of Florida, Gainesville, Florida, USA
now at: GEOMAR Helmholtz Centre for Ocean Research Kiel, Kiel, Germany
Wolfgang Bangerth
Department of Mathematics, Colorado State University, Fort Collins, Colorado, USA
Elbridge Gerry Puckett
Department of Mathematics, University of California, Davis, California, USA
Cedric Thieulot
Department of Earth Sciences, Utrecht University, Utrecht, the Netherlands
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Cited articles
Adamuszek, M., Dabrowski, M., and Schmid, D. W.: Folder: A numerical tool to simulate the development of structures in layered media, J. Struct. Geol., 84, 85–101, 2016. a
Arndt, D., Fehn, N., Kanschat, G., Kormann, K., Kronbichler, M., Munch, P., Wall, W. A., and Witte, J.: ExaDG: High-Order Discontinuous Galerkin for the Exa-Scale, in: Software for Exascale Computing – SPPEXA 2016-2019, edited by: Bungartz, H.-J., Reiz, S., Uekermann, B., Neumann, P., and Nagel, W. E., Springer International Publishing, Cham, 189–224, https://doi.org/10.1007/978-3-030-47956-5_8, 2020. a
Arndt, D., Bangerth, W., Bergbauer, M., Feder, M., Fehling, M., Heinz, J., Heister, T., Heltai, L., Kronbichler, M., Maier, M., Munch, P., Pelteret, J.-P., Turcksin, B., Wells, D., and Zampini, S.: The
deal.II Library, Version 9.5, J. Numer. Math., 31, 231–246, https://doi.org/10.1515/jnma-2023-0089, 2023. a
Arnould, M., Coltice, N., Flament, N., and Mallard, C.: Plate tectonics and mantle controls on plume dynamics, Earth Planet. Sc. Lett., 547, 116439, https://doi.org/10.1016/j.epsl.2020.116439, 2020. a
Baes, M., Sobolev, S., Gerya, T., and Brune, S.: Plume-Induced Subduction Initiation: Single-Slab or Multi-Slab Subduction?, Geochem. Geophy. Geosys., 21, e2019GC008663, https://doi.org/10.3389/feart.2021.766604, 2020. a
Bangerth, W., Dannberg, J., Fraters, M., Gassmoeller, R., Glerum, A., Heister, T., Myhill, R., and Naliboff, J.: ASPECT: Advanced Solver for Problems in Earth’s ConvecTion, Figshare [data set], https://doi.org/10.6084/m9.figshare.4865333, 2023. a
Bangerth, W., Dannberg, J., Fraters, M., Gassmoeller, R., Glerum, A., Heister, T., Myhill, R., and Naliboff, J.: ASPECT v2.5.0, Zenodo [code], https://doi.org/10.5281/zenodo.8200213, 2023. a
Bercovici, D. and Ricard, Y.: Plate tectonics, damage and inheritance, Nature, 508, 513–516, 2014. a
Billen, M. I.: Modeling the dynamics of subducting slabs, Annu. Rev. Earth Pl. Sci., 36, 325–356, 2008. a
Brandenburg, J., Hauri, E. H., van Keken, P. E., and Ballentine, C. J.: A multiple-system study of the geochemical evolution of the mantle with force-balanced plates and thermochemical effects, Earth Planet. Sc. Lett., 276, 1–13, 2008. a
Clevenger, T. C. and Heister, T.: Comparison between algebraic and matrix-free geometric multigrid for a Stokes problem on adaptive meshes with variable viscosity, Numerical Linear Algebr., 28, e2375, https://doi.org/10.1002/nla.2375, 2021. a
Dannberg, J. and Gassmöller, R.: Chemical trends in ocean islands explained by plume–slab interaction, P. Natl. Acad. Sci. USA, 115, 4351–4356, 2018. a
Dannberg, J., Eilon, Z., Faul, U., Gassmöller, R., Moulik, P., and Myhill, R.: The importance of grain size to mantle dynamics and seismological observations, Geochem. Geophy. Geosy., 18, 3034–3061, 2017. a
Duretz, T., May, D. A., Gerya, T. V., and Tackley, P. J.: Discretization errors and free surface stabilization in the finite difference and marker-in-cell method for applied geodynamics: A numerical study, Geochem. Geophy. Geosy., 12, Q07004, https://doi.org/10.1029/2011GC003567, 2011. a
Eilon, Z. C., Zhang, L., Gaherty, J. B., Forsyth, D. W., and Russell, J. B.: Sub-Lithospheric Small-Scale Convection Tomographically Imaged Beneath the Pacific Plate, Geophys. Res. Lett., 49, e2022GL100351, https://doi.org/10.1029/2022GL100351, 2022. a
El Geitani, T., Golshan, S., and Blais, B.: Toward High-Order CFD-DEM: Development and Validation, Industrial & Engineering Chemistry Research, 62, 1141–1159, https://doi.org/10.1021/acs.iecr.2c03546, 2023. a
Faccenna, C., Oncken, O., Holt, A. F., and Becker, T. W.: Initiation of the Andean orogeny by lower mantle subduction, Earth Planet. Sci. Lett., 463, 189–201, 2017. a
Farnetani, C. G. and Richards, M. A.: Numerical investigations of the mantle plume initiation model for flood basalt events, J. Geophys. Res.-Sol. Ea., 99, 13813–13833, 1994. a
Furuichi, M. and May, D. A.: Implicit solution of the material transport in Stokes flow simulation: Toward thermal convection simulation surrounded by free surface, Comput. Phys. Commun., 192, 1–11, 2015. a
Gassmöller, R.: Benchmarking the accuracy of higher order particle methods in geodynamic models of transient flow: Data, Zenodo [data set], https://doi.org/10.5281/zenodo.10805269, 2024. a
Glerum, A., Brune, S., Stamps, D. S., and Strecker, M. R.: Victoria continental microplate dynamics controlled by the lithospheric strength distribution of the East African Rift, Nat. Commun., 11, 1–15, 2020. a
Golshan, S. and Blais, B.: Load-Balancing Strategies in Discrete Element Method Simulations, Processes, 10, 79, https://doi.org/10.3390/pr10010079, 2022. a
Golshan, S., Munch, P., Gassmöller, R., Kronbichler, M., and Blais, B.: Lethe-DEM: An open-source parallel discrete element solver with load balancing, Computational Particle Mechanics, 1–20, https://doi.org/10.1007/s40571-022-00478-6, 2022. a
Grima, A. G., Lithgow-Bertelloni, C., and Crameri, F.: Orphaning regimes: the missing link between flattened and penetrating slab morphologies, Front. Earth Sci., 8, 374, https://doi.org/10.3389/feart.2020.00374, 2020. a
Guermond, J.-L., Pasquetti, R., and Popov, B.: Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230, 4248–4267, 2011. a
Gülcher, A. J. P., Ballmer, M. D., and Tackley, P. J.: Coupled dynamics and evolution of primordial and recycled heterogeneity in Earth's lower mantle, Solid Earth, 12, 2087–2107, https://doi.org/10.5194/se-12-2087-2021, 2021. a
Gurnis, M. and Hager, B. H.: Controls of the structure of subducted slabs, Nature, 335, 317–321, 1988. a
Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, ISBN-10: 3540604529, ISBN-13: 978-3540604525, 1991. a
He, Y., Puckett, E. G., and Billen, M. I.: A Discontinuous Galerkin Method with a Bound Preserving Limiter for the Advection of non-Diffusive Fields in Solid Earth Geodynamics, Phys. Earth Planet. Int., 263, 23–37, https://doi.org/10.1016/j.pepi.2016.12.001, 2016. a
Huang, J. and Zhong, S.: Sublithospheric small-scale convection and its implications for the residual topography at old ocean basins and the plate model, J. Geophys. Res.-Sol. Ea., 110, B05404, https://doi.org/10.1029/2004JB003153, 2005. a
Jones, T. D., Sime, N., and van Keken, P.: Burying Earth's primitive mantle in the slab graveyard, Geochem. Geophy. Geosy., 22, e2020GC009396, https://doi.org/10.1029/2020GC009396, 2021. a
Kaus, B. J., Popov, A. A., Baumann, T., Pusok, A., Bauville, A., Fernandez, N., and Collignon, M.: Forward and inverse modelling of lithospheric deformation on geological timescales, in: Proceedings of nic symposium, John von Neumann Institute for Computing (NIC), NIC Series, 8, 978–983, https://hdl.handle.net/2128/10411 (last access: 9 May 2024) 2016. a
Kellogg, L. and Turcotte, D.: Mixing and the distribution of heterogeneities in a chaotically convecting mantle, J. Geophys. Res.-Sol. Ea., 95, 421–432, 1990. a
Kramer, S. C., Davies, D. R., and Wilson, C. R.: Analytical solutions for mantle flow in cylindrical and spherical shells, Geosci. Model Dev., 14, 1899–1919, https://doi.org/10.5194/gmd-14-1899-2021, 2021. a
Lin, S.-C. and van Keken, P. E.: Dynamics of thermochemical plumes: 1. Plume formation and entrainment of a dense layer, Geochem. Geophy. Geosy., 7, Q02006, https://doi.org/10.1029/2005GC001071, 2006. a
McNamara, A. K. and Zhong, S.: Thermochemical structures within a spherical mantle: Superplumes or piles?, J. Geophys. Res., 109, 1–14, https://doi.org/10.1029/2003JB002847, 2004. a, b
McNamara, A. K. and Zhong, S.: Thermochemical structures beneath Africa and the Pacific Ocean, Nature, 437, 1136–1139, 2005. a
Moresi, L., Zhong, S., Han, L., Conrad, C., Tan, E., Gurnis, M., Choi, E., Thoutireddy, P., Manea, V., McNamara, A., Becker, T., Leng, W., and Armendariz, L.: CitcomS v3.3.1, Zenodo [code], https://doi.org/10.5281/zenodo.7271920, 2022. a
Murer, M., Formica, G., Milicchio, F., Morganti, S., and Auricchio, F.: A coupled multiphase Lagrangian-Eulerian fluid-dynamics framework for numerical simulation of Laser Metal Deposition process, The International Journal of Advanced Manufacturing Technology, 120, 3269–3286, https://doi.org/10.1007/s00170-022-08763-7, 2022. a
Neuharth, D., Brune, S., Glerum, A., Heine, C., and Welford, J. K.: Formation of continental microplates through rift linkage: Numerical modeling and its application to the Flemish Cap and Sao Paulo Plateau, Geochem. Geophy. Geosy., 22, e2020GC009615, https://doi.org/10.1029/2020GC009615, 2021. a
Popov, A. and Marchevsky, I.: MPI-Based PFEM-2 Method Solver for Convection-Dominated CFD Problems, in: International Conference on Parallel Computational Technologies, Springer, 261–275, https://doi.org/10.1007/978-3-031-11623-0_18, 2022. a
Puckett, E. G., Turcotte, D. L., He, Y., Lokavarapu, H., Robey, J. M., and Kellogg, L. H.: New numerical approaches for modeling thermochemical convection in a compositionally stratified fluid, Phys. Earth Planet. Int., 276, 10–35, 2018. a
Pusok, A. E., Kaus, B. J., and Popov, A. A.: On the quality of velocity interpolation schemes for marker-in-cell method and staggered grids, Pure Appl. Geophys., 174, 1071–1089, 2017. a
Richards, F., Hoggard, M., Cowton, L., and White, N.: Reassessing the thermal structure of oceanic lithosphere with revised global inventories of basement depths and heat flow measurements, J. Geophys. Res.-Sol. Ea., 123, 9136–9161, 2018. a
Samuel, H.: A deformable particle-in-cell method for advective transport in geodynamic modelling, Geophys. J. Int., 214, 1744–1773, 2018. a
Schierjott, J. C., Thielmann, M., Rozel, A. B., Golabek, G. J., and Gerya, T. V.: Can grain size reduction initiate transform faults? Insights from a 3-D numerical study, Tectonics, 39, e2019TC005793, https://doi.org/10.1029/2019TC005793, 2020. a
Schmid, D. W. and Podladchikov, Y. Y.: Analytical solutions for deformable elliptical inclusions in general shear, Geophys. J. Int., 155, 269–288, 2003. a
Schubert, G., Turcotte, D. L., and Olson, P.: Mantle Convection in the Earth and Planets, Part 1, Cambridge, ISBN13: 978-0521353670, 2001. a
Sime, N., Maljaars, J. M., Wilson, C. R., and van Keken, P. E.: An exactly mass conserving and pointwise divergence free velocity method: Application to compositional buoyancy driven flow problems in geodynamics, Geochem. Geophy. Geosy., 22, e2020GC009349, https://doi.org/10.1029/2020GC009349, 2021. a, b, c
Sime, N., Wilson, C. R., and van Keken, P. E.: A pointwise conservative method for thermochemical convection under the compressible anelastic liquid approximation, Geochem. Geophy. Geosy., 23, e2021GC009922, https://doi.org/10.1029/2021GC009922, 2022. a
Stein, C. A. and Stein, S.: A model for the global variation in oceanic depth and heat flow with lithospheric age, Nature, 359, 123–129, 1992. a
Strauss, W. A.: Partial differential equations: An introduction, John Wiley & Sons, ISBN13 978-0470054567, 2007. a
Tackley, P. J.: Self-consistent generation of tectonic plates in three-dimensional mantle convection, Earth Planet. Sc. Lett., 157, 9–22, 1998. a
Tackley, P. J.: Modelling compressible mantle convection with large viscosity contrasts in a three-dimensional spherical shell using the yin-yang grid, Phys. Earth Planet. Int., 171, 7–18, 2008. a
Tackley, P. J. and King, S. D.: Testing the tracer ratio method for modeling active compositional fields in mantle convection simulations, Geochem. Geophy. Geosy., 4, 8302, https://doi.org/10.1029/2001GC000214, 2003. a, b, c
Thieulot, C. and Bangerth, W.: On the choice of finite element for applications in geodynamics, Solid Earth, 13, 229–249, https://doi.org/10.5194/se-13-229-2022, 2022. a
Trim, S. J., Butler, S. L., McAdam, S. S., and Spiteri, R. J.: Manufacturing an exact solution for 2D thermochemical mantle convection models, Geochem. Geophy. Geosy., 24, e2022GC010807, https://doi.org/10.1029/2022GC010807, 2023a. a, b
Trim, S. J., Butler, S. L., and Spiteri, R. J.: The impact of velocity update frequency on time accuracy for mantle convection particle methods, Authorea Preprints, https://doi.org/10.22541/essoar.169444235.56582698/v1, 2023b. a
Van Dinther, Y., Gerya, T., Dalguer, L., Mai, P. M., Morra, G., and Giardini, D.: The seismic cycle at subduction thrusts: Insights from seismo-thermo-mechanical models, J. Geophys. Res.-Sol. Ea., 118, 6183–6202, 2013. a
Van Zelst, I., Wollherr, S., Gabriel, A.-A., Madden, E. H., and van Dinther, Y.: Modeling megathrust earthquakes across scales: One-way coupling from geodynamics and seismic cycles to dynamic rupture, J. Geophys. Res.-Sol. Ea., 124, 11414–11446, 2019. a
Wang, H., Agrusta, R., and van Hunen, J.: Advantages of a conservative velocity interpolation (CVI) scheme for particle-in-cell methods with application in geodynamic modeling, Geochem. Geophy. Geosy., 16, 2015–2023, https://doi.org/10.1002/2015GC005824, 2015. a
Zhong, S.: Analytic solutions for Stokes' flow with lateral variations in viscosity, Geophys. J. Int., 124, 18–28, 1996. a
Zhong, S. and Hager, B. H.: Entrainment of a dense layer by thermal plumes, Geophys. J. Int., 154, 666–676, 2003. a
Zhong, S., McNamara, A., Tan, E., Moresi, L., and Gurnis, M.: A benchmark study on mantle convection in a 3-D spherical shell using CitcomS, Geochem. Geophy. Geosy., 9, Q10017, https://doi.org/10.1029/2008GC002048, 2008. a, b
Short summary
Numerical models that use simulated particles are a powerful tool for investigating flow in the interior of the Earth, but the accuracy of these models is not fully understood. Here we present two new benchmarks that allow measurement of model accuracy. We then document that better accuracy matters for applications like convection beneath an oceanic plate. Our benchmarks and methods are freely available to help the community develop better models.
Numerical models that use simulated particles are a powerful tool for investigating flow in the...
