DOI: https://doi.org/10.26089/NumMet.v16r335

A parallel implementation of a fast method for solving the Smoluchowski-type kinetic equations of aggregation and fragmentation processes

Authors

  • S.A. Matveev

Keywords:

Smoluchowski equation
kinetics of aggregation and fragmentation processes
predictor-corrector scheme
cross interpolation method
low-rank matrix approximations
discrete convolution
parallel computing

Abstract

A parallel implementation of a fast algorithm for solving systems of the Smoluchowski-type kinetic equations of aggregation and fragmentation processes is proposed. The efficiency and scalability of the proposed implementation are shown for several particular problems of aggregation and fragmentation kinetics. The oscillatory solutions of the Cauchy problems are found using the developed parallel algorithm in terms of total density.

New computational technologies

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Published

2015-07-05

Issue

Vol. 16 (2015): Issue 3.

Section

Section 1. Numerical methods and applications

Author Biography

S.A. Matveev

Lomonosov Moscow State University,
Faculty of Computational Mathematics and Cybernetics
• Student

References

  1. N. V. Brilliantov, A. S. Bodrova, and P. L. Krapivsky, “A Model of Ballistic Aggregation and Fragmentation,” J. Stat. Mech. Theor. Exp. (2009).
    doi 10.1088/1742-5468/2009/06/P06011
  2. F. Leyvraz, “Scaling Theory and Exactly Solved Models in the Kinetics of Irreversible Aggregation,” Physics Reports 383 (2-3), 95-212 (2003).
  3. P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge Univ. Press, Cambridge, 2010).
  4. T. Pöschel, N. V. Brilliantov, and C. Frömmel, “Kinetics of Prion Growth,” Biophys. J. 85 (6), 3460-3474 (2003).
  5. J. N. Cuzzi, J. A. Burns, S. Charnoz, et al., “An Evolving View of Saturn’s Dynamic Rings,” Science 327 (5972), 1470-1475 (2010).
  6. N. Brilliantov, P. Krapivsky, A. Bodrova, et al., “Particle Size Distribution in Saturn’s Rings: Aggregation-Fragmentation Model,” in Proc. National Academy of Sciences of the United States of America, 2014 , submitted.
  7. V. A. Kazeev, B. N. Khoromskij, and E. E. Tyrtyshnikov, “Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity,” SIAM J. Sci. Comp. 35 (3), A1511-A1536 (2013).
  8. V. A. Galkin, The Smoluchowski Equation (Fizmatlit, Moscow, 2001) [in Russian].
  9. A. E. Aloyan, Dynamics and Kinematics of Gas Impurities and Aerosols in the Atmosphere (Inst. of Numerical Mathematics, Russian Academy of Sciences, 2002) [in Russian].
  10. I. Oseledets and E. Tyrtyshnikov, “TT-Cross Approximation for Multidimensional Arrays,” Linear Algebra Appl. 432 (1), 70-88 (2010).
  11. F. E. Kruis, A. Maisels, and H. Fissan, “Direct Simulation Monte Carlo Method for Particle Coagulation and Aggregation,” AIChE J. 46 (9), 1735-1742 (2000).
  12. G. Palaniswaamy and S. K. Loyalka, “Direct Simulation Monte Carlo Aerosol Dynamics: Coagulation and Collisional Sampling,” Nucl. Technol. 156 (1), 29-38 (2006).
  13. R. C. Ball, C. Connaughton, P. P. Jones, et al., “Collective Oscillations in Irreversible Coagulation Driven by Monomer Inputs and Large-Cluster Outputs,” Phys. Rev. Lett. 109 (2012).
    doi 10.1103/PhysRevLett.109.168304
  14. S. A. Matveev, E. E. Tyrtyshnikov, A. P. Smirnov, and N. V. Brilliantov, “A Fast Numerical Method for Solving the Smoluchowski-Type Kinetic Equations of Aggregation and Fragmentation Processes,” Vychisl. Metody Programm. 15, 1-8 (2014).
  15. D. A. Zheltkov and E. E. Tyrtyshnikov, “A Parallel Implementation of the Matrix Cross Approximation Method,” Vychisl. Metody Programm. 16, 369-375 (2015).
  16. S. A. Matveev, A. P. Smirnov, and E. E. Tyrtyshnikov, “A Fast Numerical Method for the Cauchy Problem for the Smoluchowski Equation,” J. Comput. Phys. 282, 23-32 (2015).
  17. E. Tyrtyshnikov, “Mosaic-Skeleton Approximations,” Calcolo 33 (1-2), 47-57 (1996).
  18. A. Gupta and V. Kumar, “The Scalability of FFT on Parallel Computers,” IEEE Trans. Parallel Distrib. Syst. 4 (8), 922-932 (1993).