An efficient finite-difference method for solving Smoluchowski-type kinetic equations of aggregation with three-body collisions
DOI:
https://doi.org/10.26089/NumMet.v19r325Keywords:
three-body Smoluchowski equation, kinetics of aggregation processes, predictor-corrector scheme, low-rank tensor approximations, discrete convolutionAbstract
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.
References
- M. Smoluchowski, “Versuch Einer Mathematischen Theorie der Koagulationskinetik Kolloider Lösungen,” Z. Phys. Chem. 92, 129-168 (1918).
- Z. A. Melzak, “A Scalar Transport Equation,” Trans. Am. Math. Soc. 1957. 85 (2), 547-560 (1957).
- V. A. Galkin, Smoluchowski equation (Fizmatlit, Moscow, 2001) [in Russian].
- F. Leyvraz, “Scaling Theory and Exactly Solved Models in the Kinetics of Irreversible Aggregation,” Phys. Rep. 383 (2-3), 95-212 (2003).
- 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).
- 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).
- G. Palaniswaamy and S. K. Loyalka, “Direct Simulation Monte Carlo Aerosol Dynamics: Coagulation and Collisional Sampling,” Nucl. Technol. 156 (1), 29-38 (2006).
- 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).
- 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).
- 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).
- P. L. Krapivsky, “Aggregation Processes with n-Particle Elementary Reactions,” J. Phys. A: Math. Gen. 24 (19), 4697-4703 (1991).
- P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge Univ. Press, Cambridge, 2010).
- Y. Jiang and H. Gang, “Generalized Smoluchovski Equation with Gelation,” Phys. Rev. B: Condens. Matter 39 (7), 4659-4665 (1989).
- 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).
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2018 Вычислительные методы и программирование
This work is licensed under a Creative Commons Attribution 4.0 International License.
