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

  • Anna WawrzynczakEmail author
  • Renata Modzelewska
  • Agnieszka Gil
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10777)


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.


Numerical algorithms Fokker-Planck equation Stochastic differential equations Cosmic ray transport Julia parallel programming 



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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Anna Wawrzynczak
    • 1
    Email author
  • Renata Modzelewska
    • 2
  • Agnieszka Gil
    • 2
  1. 1.Institute of Computer SciencesSiedlce UniversitySiedlcePoland
  2. 2.Institute of Mathematics and PhysicsSiedlce UniversitySiedlcePoland

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