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