Advertisement

Applied Mathematics and Mechanics

, Volume 39, Issue 1, pp 83–102 | Cite as

Stable and accurate schemes for smoothed dissipative particle dynamics

  • G. FaureEmail author
  • G. Stoltz
Article
  • 106 Downloads

Abstract

Smoothed dissipative particle dynamics (SDPD) is a mesoscopic particle method that allows to select the level of resolution at which a fluid is simulated. The numerical integration of its equations of motion still suffers from the lack of numerical schemes satisfying all the desired properties such as energy conservation and stability. Similarities between SDPD and dissipative particle dynamics with energy (DPDE) conservation, which is another coarse-grained model, enable adaptation of recent numerical schemes developed for DPDE to the SDPD setting. In this article, a Metropolis step in the integration of the fluctuation/dissipation part of SDPD is introduced to improve its stability.

Key words

smoothed dissipative particle dynamics (SDPD) numerical integration Metropolis algorithm 

Chinese Library Classification

O357.3 

2010 Mathematics Subject Classification

82-08 65C30 82C80 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We thank J. B. MAILLET for helpful discussion.

References

  1. [1]
    Frenkel, D. and Smit, B. Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, New York (2001)zbMATHGoogle Scholar
  2. [2]
    Tuckerman, M. Statistical Mechanics: Theory and Molecular Simulation, Oxford University Press, Oxford (2010)zbMATHGoogle Scholar
  3. [3]
    Leimkuhler, B. and Matthews, C. Molecular Dynamics: With Deterministic and Stochastic Numerical Methods, Springer, New York (2015)zbMATHGoogle Scholar
  4. [4]
    Espa˜nol, P. and Revenga, M. Smoothed dissipative particle dynamics. Physical Review E, 67, 026705 (2003)CrossRefGoogle Scholar
  5. [5]
    Lucy, L. B. A numerical approach to the testing of the fission hypothesis. Astronomical Journal, 82, 1013–1024 (1977)CrossRefGoogle Scholar
  6. [6]
    Gingold, R. A. and Monaghan, J. J. Smoothed particle hydrodynamics—–theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society, 181, 375–389 (1977)CrossRefzbMATHGoogle Scholar
  7. [7]
    Avalos, J. B. and Mackie, A. D. Dissipative particle dynamics with energy conservation. Europhysics Letters, 40, 141–146 (1997)CrossRefGoogle Scholar
  8. [8]
    Espa˜nol, P. Dissipative particle dynamics with energy conservation. Europhysics Letters, 40, 631–636 (1997)CrossRefGoogle Scholar
  9. [9]
    Faure, G., Maillet, J. B., Roussel, J., and Stoltz, G. Size consistency in smoothed dissipative particle dynamics. Physical Review E, 94, 043305 (2016)CrossRefGoogle Scholar
  10. [10]
    Vázquez-Quesada, A., Ellero, M., and Espa˜nol, P. Consistent scaling of thermal fluctuations in smoothed dissipative particle dynamics. Journal of Chemical Physics, 130, 034901 (2009)CrossRefGoogle Scholar
  11. [11]
    Litvinov, S., Ellero, M., Hu, X., and Adams, N. A. Self-diffusion coefficient in smoothed dissipative particle dynamics. Journal of Chemical Physics, 130, 021101 (2009)CrossRefGoogle Scholar
  12. [12]
    Bian, X., Litvinov, S., Qian, R., Ellero, M., and Adams, N. A. Multiscale modeling of particle in suspension with smoothed dissipative particle dynamics. Physics of Fluids, 24, 012002 (2012)CrossRefGoogle Scholar
  13. [13]
    Litvinov, S., Ellero, M., Hu, X., and Adams, N. A. Smoothed dissipative particle dynamics model for polymer molecules in suspension. Physical Review E, 77, 066703 (2008)CrossRefGoogle Scholar
  14. [14]
    Petsev, N. D., Leal, L. G., and Shell, M. S. Multiscale simulation of ideal mixtures using smoothed dissipative particle dynamics. Journal of Chemical Physics, 144, 084115 (2016)CrossRefGoogle Scholar
  15. [15]
    Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10, 545–551 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Strang, G. On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis, 5, 506–517 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Verlet, L. Computer “experiments” on classical fluids I: thermodynamical properties of Lennard- Jones molecules. Physical Review, 159, 98–103 (1967)CrossRefGoogle Scholar
  18. [18]
    Hoogerbrugge, P. J. and Koelman, J. M. V. A. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhysics Letters, 19, 155–160 (1992)CrossRefGoogle Scholar
  19. [19]
    Stoltz, G. A reduced model for shock and detonation waves I: the inert case. Europhysics Letters, 76, 849–855 (2006)MathSciNetCrossRefGoogle Scholar
  20. [20]
    Shardlow, T. Splitting for dissipative particle dynamics. SIAM Journal on Scientific Computing, 24, 1267–1282 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    L´ısal, M., Brennan, J. K., and Avalos, J. B. Dissipative particle dynamics at isothermal, isobaric, isoenergetic, and isoenthalpic conditions using Shardlow-like splitting algorithms. Journal of Chemical Physics, 135, 204105 (2011)CrossRefGoogle Scholar
  22. [22]
    Larentzos, J. P., Brennan, J. K., Moore, J. D., L´ısal, M., and Mattson, W. D. Parallel implementation of isothermal and isoenergetic dissipative particle dynamics using Shardlow-like splitting algorithms. Computer Physics Communications, 185, 1987–1998 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Homman, A. A., Maillet, J. B., Roussel, J., and Stoltz, G. New parallelizable schemes for integrating the dissipative particle dynamics with energy conservation. Journal of Chemical Physics, 144, 024112 (2016)CrossRefGoogle Scholar
  24. [24]
    Langenberg, M. and Müller M. eMC: a Monte Carlo scheme with energy conservation. Europhysics Letters, 114, 20001 (2016)CrossRefGoogle Scholar
  25. [25]
    Stoltz, G. Stable schemes for dissipative particle dynamics with conserved energy. Journal of Computational Physics, 340, 451–469 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Litvinov, S., Ellero, M., Hu, X., and Adams, N. A splitting scheme for highly dissipative smoothed particle dynamics. Journal of Computational Physics, 229, 5457–5464 (2010)CrossRefzbMATHGoogle Scholar
  27. [27]
    Liu, G. R. and Liu, M. B. Smoothed Particle Hydrodynamics, a Meshfree Particle Method, World Scientific Publishing, Singapore (2003)CrossRefzbMATHGoogle Scholar
  28. [28]
    Liu, M., Liu, G., and Lam, K. Constructing smoothing functions in smoothed particle hydrodynamics with applications. Journal of Computational and Applied Mathematics, 155, 263–284 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Hairer, E., Lubich, C., and Wanner, G. Geometric numerical integration illustrated by the Störmer-Verlet method. Acta Numerica, 12, 399–450 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Hairer, E., Lubich, C., and Wanner, G. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  31. [31]
    Marsh, C. Theoretical Aspects of Dissipative Particle Dynamics, Ph. D. dissertation, University of Oxford, Oxford (1998)Google Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CEA, DAM, DIFArpajonFrance
  2. 2.Université Paris-Est, CERMICS (ENPC)INRIAMarne-la-ValléeFrance

Personalised recommendations