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Algorithms for Forward and Backward Solution of the Fokker-Planck Equation in the Heliospheric Transport of Cosmic Rays

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Parallel Processing and Applied Mathematics (PPAM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10777))

Abstract

Motion of charged particles in an inhomogeneous turbulent medium as magnetic field is described by partial differential equations of the Fokker-Planck-Kolmogorov type. We present an algorithm of numerical solution of the four-dimensional Fokker-Planck equation in three-dimensional spherical coordinates system. The algorithm is based on Monte Carlo simulations of the stochastic motion of quasi-particles guided by the set of stochastic differential equations corresponding to the Fokker-Planck equation by the Ito formalism. We present the parallel algorithm in Julia programming language. We simulate the transport of cosmic rays in the heliosphere considering the full three-dimensional diffusion tensor. We compare forward- and backward-in-time solutions of the transport equation and discuss its computational advantages and disadvantages.

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References

  1. Fokker, D.: Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Ann. Phys. 348, 810–820 (1914)

    Article  Google Scholar 

  2. Planck, M.: An essay on statistical dynamics and its amplification in the quantum theory. Sitzber. Preuss. Akad. Wiss. 325, 324 (1917)

    Google Scholar 

  3. Parker, E.: The passage of energetic charged particles through interplanetary space. Planet. Space Sci. 13, 9–49 (1965)

    Article  Google Scholar 

  4. Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gardiner, C.W.: Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer Series in Synergetics. Springer, Heidelberg (2009)

    MATH  Google Scholar 

  6. Moraal, H.: Cosmic-ray modulation equations. Space Sci. Rev. 176, 299–319 (2013)

    Article  Google Scholar 

  7. Gervasi, M., Rancoita, P.G., Usoskin, I.G., Kovaltsov, G.A.: Monte-Carlo approach to galactic cosmic ray propagation in the heliosphere. Nucl. Phys. B (Proc. Suppl.) 78, 26–31 (1999)

    Article  Google Scholar 

  8. Alania, M.V.: Stochastic variations of galactic cosmic rays. Acta Physica Pol. B 33(4), 1149–1166 (2002)

    Google Scholar 

  9. Zhang, M.: A markov stochastic process theory of cosmic-ray modulation. Astrophys. J. 513, 409–420 (1999)

    Article  Google Scholar 

  10. Pei, C., Bieber, J.W., Burger, R.A., Clem, J.: A general time-dependent stochastic method for solving Parker’s transport equation in spherical coordinates. J. Geophys. Res. 115, A12107 (2010)

    Google Scholar 

  11. Vos, E., Potgieter, M.S.: New modeling of galactic proton modulation during the minimum of solar cycle 23/24. Astrophys. J., 815, article id. 119, 8 pp. (2015)

    Google Scholar 

  12. Bobik, P., Boschini, M.J., Della Torre, S., et al.: On the forward-backward-in-time approach for Monte Carlo solution of Parker’s transport equation: one-dimensional case. J. Geophys. Res. Space Phys. 121, 3920–3930 (2016)

    Article  Google Scholar 

  13. Kopp, A., Busching, I., Strauss, R.D., Potgieter, M.S.: A stochastic differential equation code for multidimensional Fokker-Planck type problems. Comput. Phys. Commun. 183, 530–542 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wawrzynczak, A., Modzelewska, R., Gil, A.: Stochastic approach to the numerical solution of the non-stationary Parker’s transport equation. J. Phys. Conf. Ser. 574, 012078 (2015)

    Article  Google Scholar 

  15. Kloeden, P.E., Platen, E., Schurz, H.: Numerical Solution of SDE Through Computer Experiments. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-642-57913-4

    MATH  Google Scholar 

  16. https://julialang.org/

  17. Wawrzynczak, A., Modzelewska, R., Kluczek, M.: Numerical methods for solution of the stochastic differential equations equivalent to the non-stationary Parkers transport equation. J. Phys. Conf. Ser. 633, 012058 (2015)

    Article  Google Scholar 

  18. https://github.com/JuliaParallel/DistributedArrays.jl

  19. Wawrzynczak, A., Modzelewska, R., Gil, A.: A stochastic method of solution of the Parker transport equation. J. Phys. Conf. Ser. 632, 1742–6596 (2015)

    Google Scholar 

Download references

Acknowledgments

This work is supported by The Polish National Science Centre grant awarded by decision number DEC-2012/07/D/ST6/02488. Calculations were performed at the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM) at Warsaw University within the computational grant no. G66-19.

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Authors and Affiliations

  1. Institute of Computer Sciences, Siedlce University, Siedlce, Poland

    Anna Wawrzynczak

  2. Institute of Mathematics and Physics, Siedlce University, Siedlce, Poland

    Renata Modzelewska & Agnieszka Gil

Authors
  1. Anna Wawrzynczak
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  2. Renata Modzelewska
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  3. Agnieszka Gil
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Corresponding author

Correspondence to Anna Wawrzynczak .

Editor information

Editors and Affiliations

  1. Czestochowa University of Technology, Czestochowa, Poland

    Roman Wyrzykowski

  2. University of Tennessee, Knoxville, Tennessee, USA

    Jack Dongarra

  3. University of Southern California, Marina Del Rey, California, USA

    Ewa Deelman

  4. Czestochowa University of Technology, Czestochowa, Poland

    Konrad Karczewski

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Wawrzynczak, A., Modzelewska, R., Gil, A. (2018). Algorithms for Forward and Backward Solution of the Fokker-Planck Equation in the Heliospheric Transport of Cosmic Rays. In: Wyrzykowski, R., Dongarra, J., Deelman, E., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2017. Lecture Notes in Computer Science(), vol 10777. Springer, Cham. https://doi.org/10.1007/978-3-319-78024-5_2

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  • DOI: https://doi.org/10.1007/978-3-319-78024-5_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-78023-8

  • Online ISBN: 978-3-319-78024-5

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