Abstract

We study numerical methods for dissipative particle dynamics, a system of stochastic differential equations for simulating particles interacting pairwise according to a soft potential at constant temperature where the total momentum is conserved. We introduce splitting methods and examine the behavior of these methods experimentally. The performance of the methods, particularly temperature control, is compared to the modified velocity Verlet method used in many previous papers.

MSC codes

  1. 60H10
  2. 82C80

Keywords

  1. stochastic differential equations
  2. numerical methods
  3. molecular dynamics

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
E. S. Boek, P. V. Coveney, H. N. W. Lekkerkerker, and P. van der Schoot, Simulating the rheology of dense colloidal suspensions using dissipative particle dynamics, Phys. Rev. E (3), 55 (1997), pp. 3124–3133.
2.
A. Bruenger, C. L. Brooks III, and M. Karplus, Stochastic boundary conditions for molecular dynamics simulations of ST2 water, Chem. Phys. Lett., 105 (1984), pp. 495–500.
3.
K. Burrage, P. Burrage, High strong order methods for non‐commutative stochastic ordinary differential equation systems and the Magnus formula, Phys. D, 133 (1999), 34–48, Predictability: quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998)
4.
P. Coveney and K. E. Novick, Computer simulations of domain growth and phase separation in two dimensional binary immiscible fluids using dissipative particle dynamics, Phys. Rev. E (3), (1996), pp. 5134–5141.
5.
P. Espanol and P. Warren, Statistical mechanics of dissipative particle dynamics, Europhys. Lett., 30 (1995), pp. 191–196.
6.
J. B. Gibson, K. Chen, and S. Chynoweth, The equilibrium of a velocity‐Verlet type algorithm for DPD with finite time steps, Internat. J. Modern Phys. C, 10 (1999), pp. 241–261.
7.
R. D. Groot and T. Madden, Dynamic simulation of diblock copolymer microphase separation, J. Chem. Phys., 108 (1998), pp. 8713–8724.
8.
R. D. Groot and P. B. Warren, Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation, J. Chem. Phys., 107 (1997), pp. 4423–4435.
9.
P. J. Hoogerbrugge and J. M. V. A. Koelman, Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics, Europhys. Lett., 19 (1992), pp. 155–160.
10.
Ioannis Karatzas, Steven Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics, Vol. 113, Springer‐Verlag, 1991xxiv+470
11.
Peter Kloeden, Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), Vol. 23, Springer‐Verlag, 1992xxxvi+632
12.
Y. Kong, C. W. Manke, W. G. Madden, and A. G. Schlijper, Effect of solvent quality on the conformation and relaxation of polymers via the dissipative particle dynamics, J. Chem. Phys., 107 (1997), pp. 592–602.
13.
C. A. Marsh and J. M. Yeomans, Dissipative particle dynamics: The equilibrium for finite time steps, Europhys. Lett., 37 (1997), pp. 511–516.
14.
Tetsuya Misawa, A Lie algebraic approach to numerical integration of stochastic differential equations, SIAM J. Sci. Comput., 23 (2001), 866–890
15.
I. Pagonabarraga, M. H. J. Hagen, and D. Frenkel, Self‐consistent dissipative particle dynamics algorithm, Europhys. Lett., 42 (1998), pp. 377–382.
16.
R. W. Pastor, B. R. Brooks, and A. Szabo, An analysis of the accuracy of Langevin and molecular dynamics algorithms, Molecular Phys., 65 (1988), pp. 1409–1419.
17.
W. Petersen, A general implicit splitting for stabilizing numerical simulations of Itô stochastic differential equations, SIAM J. Numer. Anal., 35 (1998), 1439–1451
18.
M. Revenga, I. Zuniga, and P. Espanol, Boundary model in DPD, Int. J. Mod. Phys., 9 (1998), pp. 1319–1328.
19.
J. Sanz‐Serna, M. Calvo, Numerical Hamiltonian problems, Applied Mathematics and Mathematical Computation, Vol. 7, Chapman & Hall, 1994xii+207
20.
T. Schlick, E. Barth, and M. Mandziuk, Biomolecular dynamics at long timesteps, Annu. Rev. Biophys. Biomol. Struct., 26 (1997), pp. 181–222.
21.
A. G. Schlijper, P. J. Hoogerbrugge, and C. W. Manke, Computer simulation of dilute polymer solutions with the dissipative particle dynamics method, J. Rheol., 39 (1995), pp. 567–579.
22.
T. Shardlow, Numerical Methods for DPD, Tech. Rep. NA‐01/06, University of Durham, Durham, UK, 2001.
23.
G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), pp. 506–517.
24.
P. B. Warren, Dissipative particle dynamics, Current Opinion in Colloid & Interface Science, 3 (1998), pp. 620–624.

Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: 1267 - 1282
ISSN (online): 1095-7197

History

Published online: 25 July 2006

MSC codes

  1. 60H10
  2. 82C80

Keywords

  1. stochastic differential equations
  2. numerical methods
  3. molecular dynamics

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.