Skip to main content
SpringerLink
Log in

Modifications of weighted Monte Carlo algorithms for nonlinear kinetic equations

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

Test problems for the nonlinear Boltzmann and Smoluchowski kinetic equations are used to analyze the efficiency of various versions of weighted importance modeling as applied to the evolution of multiparticle ensembles. For coagulation problems, a considerable gain in computational costs is achieved via the approximate importance modeling of the “free path” of the ensemble combined with the importance modeling of the index of a pair of interacting particles. A weighted modification of the modeling of the initial velocity distribution was found to be the most efficient for model solutions to the Boltzmann equation. The technique developed can be useful as applied to real-life coagulation and relaxation problems for which the model problems considered give approximate solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Kac, Probability and Related Topics in Physical Sciences (Interscience, London, 1959; Mir, Moscow, 1965).

    MATH  Google Scholar 

  2. G. A. Mikhailov and S. V. Rogasinsky, “Weighted Monte Carlo Methods for Approximate Solution to the Nonlinear Boltzmann Equation,” Sib. Mat. Zh. 43, 620–628 (2002).

    Google Scholar 

  3. G. A. Mikhailov, S. V. Rogasinsky, and N. M. Ureva, “Weighted Monte Carlo Method for an Approximate Solution to the Nonlinear Coagulation Equation,” Zh. Vychisl. Mat. Mat. Fiz. 46, 715–726 (2006) [Comput. Math. Math. Phys. 46, 680–690 (2006)].

    MATH  MathSciNet  Google Scholar 

  4. G. A. Mikhailov and A. V. Voitishek, Numerical Statistical Simulation (Monte Carlo Method) (Akademiya, Moscow, 2006) [in Russian].

    Google Scholar 

  5. V. S. Vladimirov and G. I. Marchuk, “A Definition of an Adjoint Operator for Nonlinear Problems,” Dokl. Akad. Nauk 372, 165–168 (2000) [Dokl. Math. 61, 438–441 (2000)].

    MathSciNet  Google Scholar 

  6. G. A. Mikhailov, “Construction of Weighted Monte Carlo Methods via an Increase in the Phase Space Dimension,” Dokl. Akad. Nauk 389, 461–464 (2003) [Dokl. Math. 67, 230–233 (2003)].

    MathSciNet  Google Scholar 

  7. V. M. Voloshchuk, Kinetic Coagulation Theory (Gidrometeoizdat, Leningrad, 1984) [in Russian].

    Google Scholar 

  8. A. V. Bobylev, “Exact Solutions to the Nonlinear Boltzmann Equation and the Theory of Relaxation of a Maxwell Gas,” Teor. Mat. Fiz. 60, 280–310 (1984).

    MathSciNet  Google Scholar 

  9. M. S. Ivanov, M. A. Korotchenko, G. A. Mikhailov, and S. V. Rogasinsky, “Global Weighted Monte Carlo Method for the Nonlinear Boltzmann Equation,” Zh. Vychisl. Mat. Mat. Fiz. 45, 1860–1870 (2005) [Comput. Math. Math. Phys. 45, 1792–1801 (2005)].

    MATH  Google Scholar 

  10. M. A. Marchenko and G. A. Mikhailov, “Distributed Monte Carlo Computations,” Avtom. Telemekh., No. 5 (2007).

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, pr. Akademika Lavrent’eva 6, 630090, Russia

    M. A. Korotchenko, G. A. Mikhailov & S. V. Rogasinsky

Authors
  1. M. A. Korotchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. A. Mikhailov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. V. Rogasinsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. A. Korotchenko.

Additional information

Original Russian Text © M.A. Korotechenko, G.A. Mikhailov, S.V. Rogasinsky, 2007, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 12, pp. 2110–2121.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Korotchenko, M.A., Mikhailov, G.A. & Rogasinsky, S.V. Modifications of weighted Monte Carlo algorithms for nonlinear kinetic equations. Comput. Math. and Math. Phys. 47, 2023–2033 (2007). https://doi.org/10.1134/S0965542507120123

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542507120123

Keywords

Search

Navigation