Skip to main content

Integrating an N-Body Problem with SDC and PFASST

  • Conference paper
  • First Online:
Domain Decomposition Methods in Science and Engineering XXI

Abstract

Vortex methods for the Navier–Stokes equations are based on a Lagrangian particle discretization, which reduces the governing equations to a first-order initial value system of ordinary differential equations for the position and vorticity of N particles. In this paper, the accuracy of solving this system by time-serial spectral deferred corrections (SDC) as well as by the time-parallel Parallel Full Approximation Scheme in Space and Time (PFASST) is investigated. PFASST is based on intertwining SDC iterations with differing resolution in a manner similar to the Parareal algorithm and uses a Full Approximation Scheme (FAS) correction to improve the accuracy of coarser SDC iterations. It is demonstrated that SDC and PFASST can generate highly accurate solutions, and the performance in terms of function evaluations required for a certain accuracy is analyzed and compared to a standard Runge–Kutta method.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Barnes, J.E., Hut, P.: A hierarchical \(\mathcal{O}(N\log N)\) force-calculation algorithm. Nature 324(6096), 446–449 (1986)

    Article  Google Scholar 

  2. Christlieb, A., Ong, B., Qiu, J.: Comments on high order integrators embedded within integral deferred correction methods. Commun. Appl. Math. Comput. Sci. 4(1), 27–56 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Christlieb, A., Macdonald, C., Ong, B.: Parallel high-order integrators. SIAM J. Sci. Comput. 32(2), 818 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cottet, G.H., Koumoutsakos, P.: Vortex Methods: Theory and Applications, 2nd edn. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  5. Cruz, F.A., Knepley, M.G., Barba, L.A.: PetFMM-A dynamically load-balancing parallel fast multipole library. Int. J. Numer. Methods Eng. 79(13), 1577–1604 (2010)

    Article  Google Scholar 

  6. Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT Numer. Math. 40(2), 241–266 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Emmett, M., Minion, M.: Toward an efficient parallel in time method for partial differential equations. Commun. Appl. Math. Comput. Sci. 7(1), 105–132 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gibbon, P., Speck, R., Karmakar, A., Arnold, L., Frings, W., Berberich, B., Reiter, D., Masek, M.: Progress in mesh-free plasma simulation with parallel tree codes. IEEE Trans. Plasma Sci. 38(9), 2367–2376 (2010). doi:10.1109/tps.2010.2055165

    Article  Google Scholar 

  10. Gibbon, P., Winkel, M., Arnold, L., Speck, R.: PEPC website. http://www.fz-juelich.de/ias/jsc/pepc (2012)

  11. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comp. Phys. 73(2), 325–348 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Huang, J., Jia, J., Minion, M.: Accelerating the convergence of spectral deferred correction methods. J. Comput. Phys. 214(2), 633–656 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jülich Supercomputing Centre: JUROPA/HPC-FF website. http://www.fz-juelich.de/jsc/juropa (2012)

  14. Layton, A.T., Minion, M.L.: Implications of the choice of quadrature nodes for picard integral deferred corrections methods for ordinary differential equations. BIT Numer. Math. 45(2), 341–373 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lions, J.L., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDE’s. C. R. Acad. Sci. Ser. I Math. 332, 661–668 (2001)

    MATH  MathSciNet  Google Scholar 

  16. Minion, M.L.: Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1(3), 471–500 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Minion, M.L.: A hybrid parareal spectral deferred corrections method. Commun. Appl. Math. Comput. Sci. 5(2), 265–301 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nievergelt, J.: Parallel methods for integrating ordinary differential equations. Commun. ACM 7(12), 731–733 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ruprecht, D., Krause, R.: Explicit parallel-in-time integration of a linear acoustic-advection system. Comput. Fluids 59, 72–83 (2012)

    Article  MathSciNet  Google Scholar 

  20. Salmon, J.K., Warren, M.S., Winckelmans, G.: Fast parallel tree codes for gravitational and fluid dynamical N-body problems. Int. J. Supercomput. Appl. 8, 129–142 (1994)

    Article  Google Scholar 

  21. Speck, R.: Generalized algebraic kernels and multipole expansions for massively parallel vortex particle methods. Ph.D. thesis, Universität Wuppertal (2011)

    Google Scholar 

  22. Speck, R., Gibbon, P., Hofmann, M.: Efficiency and scalability of the parallel Barnes-Hut tree code PEPC. In: Chapman, B., Desprez, F., Joubert, G.R., Lichnewsky, A., Peters, F.J., Priol, T. (eds.) Parallel Computing: From Multicores and GPU’s to Petascale. Advances in Parallel Computing, vol. 19, pp. 35–42. IOS Press, Amsterdam (2010)

    Google Scholar 

  23. Speck, R., Arnold, L., Gibbon, P.: Towards a petascale tree code: Scaling and efficiency of the PEPC library. J. Comput. Sci. 2, 137–142 (2011)

    Article  Google Scholar 

  24. Speck, R., Krause, R., Gibbon, P.: Parallel remeshing in tree codes for vortex particle methods. In: Bosschere, K.D., D’Hollander, E.H., Joubert, G.R., Padua, D., Peters, F., Sawyer, M. (eds.) Applications, Tools and Techniques on the Road to Exascale Computing. Advances in Parallel Computing, vol. 22, IOS Press, Amsterdam (2012)

    Google Scholar 

  25. Speck, R., Ruprecht, D., Krause, R., Emmett, M., Minion, M., Winkel, M., Gibbon, P.: A massively space-time parallel N-body solver. In: Proceedings of the SC’12 International Conference for High Performance Computing, Networking, Storage and Analysis, vol. 92, 11 pp. (2012)

    Google Scholar 

  26. van Rees, W.M., Leonard, A., Pullin, D.I., Koumoutsakos, P.: A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical flows at high Reynolds numbers. J. Comput. Phys. 230, 2794–2805 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Winkel, M., Speck, R., Hübner, H., Arnold, L., Krause, R., Gibbon, P.: A massively parallel, multi-disciplinary Barnes-Hut tree code for extreme-scale N-body simulations. Comput. Phys. Commun. 183(4), 880–889 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research is partly funded by the Swiss “High Performance and High Productivity Computing” initiative HP2C; the Director, DOE Office of Science, Office of Advanced Scientific Computing Research, Office of Mathematics, Information, and Computational Sciences, Applied Mathematical Sciences Program, under contract DE-SC0004011; and the ExtreMe Matter Institute (EMMI) in the framework of the German Helmholtz Alliance HA216. Computing resources were provided by Jülich Supercomputing Centre under project JZAM04.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Robert Speck .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Speck, R. et al. (2014). Integrating an N-Body Problem with SDC and PFASST. In: Erhel, J., Gander, M., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-05789-7_61

Download citation

Publish with us

Policies and ethics