Skip to main content
SpringerLink
Log in

A regular fast multipole method for geometric numerical integrations of Hamiltonian systems

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

The Fast Multipole Method (FMM) has been widely developed and studied for the evaluation of Coulomb energy and Coulomb forces. A major problem occurs when the FMM is applied to approximate the Coulomb energy and Coulomb energy gradient within geometric numerical integrations of Hamiltonian systems considered for solving astronomy or molecular-dynamics problems: The FMM approximation involves an approximated potential which is not regular, implying a loss of the preservation of the Hamiltonian of the system. In this paper, we present a regularization of the Fast Multipole Method in order to recover the invariance of energy. Numerical tests are given on a toy problem to confirm the gain of such a regularization of the fast method.

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. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Wiley, New York (1972)

    MATH  Google Scholar 

  2. Board, J.A., Elliot, W.S.: Fast Fourier transform accelerated fast multipole algorithm. Technical Report 94-001, Duke University, Dept of Electrical Engineering (1994)

  3. Burant, J.C., Strain, M.C., Scuseria, G.E., Frisch, M.J.: Kohn-Sham analytic energy second derivatives with the Gaussian very fast multipole method (GvFMM). Chem. Phys. Lett. 258, 45–52 (1996)

    Article  Google Scholar 

  4. Burant, J.C., Strain, M.C., Scuseria, G.E., Frisch, M.J.: Analytic energy gradients for the Gaussian very fast multipole method (GvFMM). Chem. Phys. Lett. 248, 43–49 (1996)

    Article  Google Scholar 

  5. Greengard, L., Rokhlin, V.: The rapid evaluation of potential fields in three dimensions. In: Vortex Methods. Lecture Notes in Mathematics, vol. 1360, pp. 121–141. Springer, Berlin (1988)

    Chapter  Google Scholar 

  6. Greengard, L., Rokhlin, V.: A new version of the Fast Multipole Method for the Laplace equation in the three dimensions. Acta Numer. 6, 229–269 (1997)

    Article  MathSciNet  Google Scholar 

  7. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn., vol. 31. Springer, Berlin (2006)

    MATH  Google Scholar 

  8. Izmaylov, A.F., Scuseria, G.E.: Efficient evaluation of analytic vibrational frequencies in Hartree-Fock and density functional theory for periodic nonconducting systems. J. Chem. Phys. 127, 144106 (2007)

    Article  Google Scholar 

  9. Kudin, K.N., Scuseria, G.E.: Range definitions for Gaussian-type charge distributions in fast multipole methods. J. Chem. Phys. 111(6), 2351–2356 (1999)

    Article  Google Scholar 

  10. Kvaerno, A., Leimkuhler, B.: A time-reversible, regularized, switching integrator for the n-body problem. SIAM J. Sci. Comput. 22(3), 1016–1035 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Laird, B., Leimkuhler, B.: A molecular dynamics algorithm for mixed hard-core/continuous potentials. Mol. Phys. 98, 309–316 (2000)

    Article  Google Scholar 

  12. Petersen, H.G., Soelvason, D., Perram, J.W., Smith, E.R.: The very fast multipole method. J. Chem. Phys. 101(10), 8870–8876 (1994)

    Article  Google Scholar 

  13. Shao, Y., White, C.A., Head-Gordon, M.: Efficient evaluation of the Coulomb force in density-functional theory calculations. J. Chem. Phys. 114(15), 6572–6577 (2001)

    Article  Google Scholar 

  14. Strain, M.C., Scuseria, G.E., Frisch, M.J.: Achieving linear scaling for the electronic quantum Coulomb problem. Science 271, 51–53 (1996)

    Article  Google Scholar 

  15. White, C.A., Head-Gordon, M.: Rotating around the quartic angular momentum barrier in fast multipole method calculation. J. Chem. Phys. 105(12), 5061–5067 (1996)

    Article  Google Scholar 

  16. White, C.A., Johnson, B.G., Gill, P.M.W., Head-Gordon, M.: Linear scaling density functional calculations via the continuous fast multipole method. Chem. Phys. Lett. 253, 268–278 (1996)

    Article  Google Scholar 

  17. White, C.A., Johnson, B.G., Gill, P.M.W., Head-Gordon, M.: The continuous fast multipole method. Chem. Phys. Lett. 230, 8–16 (1994)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Equipe projet IPSO, INRIA Rennes and Ecole Normale Supérieure de Cachan—Antenne de Bretagne, Avenue Robert Schumann, 35170, Bruz, France

    P. Chartier, E. Darrigrand & E. Faou

  2. IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042, Rennes cedex, France

    E. Darrigrand

Authors
  1. P. Chartier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Darrigrand
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Faou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Darrigrand.

Additional information

Communicated by Timo Eirola.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chartier, P., Darrigrand, E. & Faou, E. A regular fast multipole method for geometric numerical integrations of Hamiltonian systems. Bit Numer Math 50, 23–40 (2010). https://doi.org/10.1007/s10543-010-0248-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-010-0248-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation