An efficient finite-difference method for solving Smoluchowski-type kinetic equations of aggregation with three-body collisions


  • D.A. Stefonishin Lomonosov Moscow State University
  • S.A. Matveev Skolkovo Institute of Science and Technology
  • A.P. Smirnov Lomonosov Moscow State University
  • E.E. Tyrtyshnikov Institute of Numerical Mathematics of RAS (INM RAS)



three-body Smoluchowski equation, kinetics of aggregation processes, predictor-corrector scheme, low-rank tensor approximations, discrete convolution


We consider a model of aggregation processes for the Smoluchowski-type kinetic equations with three-body collisions of particles. We propose a numerical method for the fast solving of Cauchy problems for the corresponding systems of equations. The proposed method allows one to reduce the step complexity O(N3) of the finite-difference predictor-corrector scheme to O(RNlogN) without loss of accuracy. Here the parameter N specifies the number of considered equations and R is the rank of kinetic coefficient arrays. The efficiency and accuracy of the proposed numerical method are demonstrated for model problems of aggregation kinetics.

Author Biographies

D.A. Stefonishin

S.A. Matveev

A.P. Smirnov

Lomonosov Moscow State University
• Associate Professor

E.E. Tyrtyshnikov


  1. M. Smoluchowski, “Versuch Einer Mathematischen Theorie der Koagulationskinetik Kolloider Lösungen,” Z. Phys. Chem. 92, 129-168 (1918).
  2. Z. A. Melzak, “A Scalar Transport Equation,” Trans. Am. Math. Soc. 1957. 85 (2), 547-560 (1957).
  3. V. A. Galkin, Smoluchowski equation (Fizmatlit, Moscow, 2001) [in Russian].
  4. F. Leyvraz, “Scaling Theory and Exactly Solved Models in the Kinetics of Irreversible Aggregation,” Phys. Rep. 383 (2-3), 95-212 (2003).
  5. A. A. Sorokin, V. F. Strizhov, M. N. Demin, and A. P. Smirnov, “Monte-Carlo Modeling of Aerosol Kinetics,” Atomic Energy 117 (4), 289-293 (2015).
  6. 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).
  7. G. Palaniswaamy and S. K. Loyalka, “Direct Simulation Monte Carlo Aerosol Dynamics: Coagulation and Collisional Sampling,” Nucl. Technol. 156 (1), 29-38 (2006).
  8. 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).
  9. 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).
  10. A. P. Smirnov, S. A. Matveev, D. A. Zheltkov, and E. E. Tyrtyshnikov, “Fast and Accurate Finite-Difference Method Solving Multicomponent Smoluchowski Coagulation Equation with Source and Sink Terms,” Procedia Comput. Sci. 80, 2141-2146 (2016).
  11. P. L. Krapivsky, “Aggregation Processes with n-Particle Elementary Reactions,” J. Phys. A: Math. Gen. 24 (19), 4697-4703 (1991).
  12. P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge Univ. Press, Cambridge, 2010).
  13. Y. Jiang and H. Gang, “Generalized Smoluchovski Equation with Gelation,” Phys. Rev. B: Condens. Matter 39 (7), 4659-4665 (1989).
  14. 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. Comput. 35 (3), A1511-A1536 (2013).



How to Cite

Стефонишин Д., Матвеев С., Смирнов А., Тыртышников Е. An Efficient Finite-Difference Method for Solving Smoluchowski-Type Kinetic Equations of Aggregation With Three-Body Collisions // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2018. 19. 261-269. doi 10.26089/NumMet.v19r325



Section 1. Numerical methods and applications

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