Numerical methods for solution of the stochastic differential equations equivalent to the non-stationary Parker's transport equation

TL;DR

This paper develops and compares advanced numerical schemes for solving stochastic differential equations derived from the non-stationary Parker transport equation, which models galactic cosmic ray transport in the heliosphere.

Contribution

It introduces strong order numerical methods, including Euler-Maruyama, Milstein, and stochastic Runge-Kutta, for solving high-dimensional SDEs related to cosmic ray transport.

Findings

Milstein and Runge-Kutta methods improve accuracy over Euler-Maruyama.

The methods handle full 3D diffusion tensors effectively.

Discussion of advantages and disadvantages guides method selection.

Abstract

We derive the numerical schemes for the strong order integration of the set of the stochastic differential equations (SDEs) corresponding to the non-stationary Parker transport equation (PTE). PTE is 5-dimensional (3 spatial coordinates, particles energy and time) Fokker- Planck type equation describing the non-stationary the galactic cosmic ray (GCR) particles transport in the heliosphere. We present the formulas for the numerical solution of the obtained set of SDEs driven by a Wiener process in the case of the full three-dimensional diffusion tensor. We introduce the solution applying the strong order Euler-Maruyama, Milstein and stochastic Runge-Kutta methods. We discuss the advantages and disadvantages of the presented numerical methods in the context of increasing the accuracy of the solution of the PTE.