Stochastic approach to the numerical solution of the non-stationary Parker's transport equation

TL;DR

This paper introduces a stochastic numerical method for solving Parker's transport equation, modeling galactic cosmic ray transport in the heliosphere, and demonstrates its effectiveness through comparison with experimental data.

Contribution

The paper develops a stochastic approach using SDEs for solving the non-stationary Parker's transport equation, providing a new computational tool for cosmic ray transport modeling.

Findings

The stochastic model agrees with experimental data on GCR intensity variations.

The method effectively handles the non-stationary 4D Parker's equation.

Discussion of forward and backward solution advantages and disadvantages.

Abstract

We present the newly developed stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. Mathematically Parker transport equation (PTE) describing non-stationary transport of charged particles in the turbulent medium is the Fokker-Planck type. It is the second order parabolic time-dependent 4-dimensional (3 spatial coordinates and particles energy/rigidity) partial differential equation. It is worth to mention that, if we assume the stationary case it remains as the 3-D parabolic type problem with respect to the particles rigidity R. If we fix the energy it still remains as the 3-D parabolic type problem with respect to time. The proposed method of numerical solution is based on the solution of the system of stochastic differential equations (SDEs) being equivalent to the Parker's transport equation. We present the method of deriving from PTE the equivalent SDEs in the heliocentric spherical coordinate system for the backward approach. The obtained stochastic model of the Forbush decrease of the GCR intensity is in an agreement with the experimental data. The advantages and disadvantages of the forward and the backward solution of the PTE are discussed.