A general time-dependent stochastic method for solving Parker's transport equation in spherical coordinates

We present a detailed description of our newly developed stochastic approach for solving Parker's transport equation, which we believe is the first attempt to solve it with time dependence in 3-D, evolving from our 3-D steady state stochastic approach. Our formulation of this method is general and is valid for any type of heliospheric magnetic field, although we choose the standard Parker field as an example to illustrate the steps to calculate the transport of galactic cosmic rays. Our 3-D stochastic method is different from other stochastic approaches in the literature in several ways. For example, we employ spherical coordinates to integrate directly, which makes the code much more efficient by reducing coordinate transformations. What is more, the equivalence between our stochastic differential equations and Parker's transport equation is guaranteed by Ito's theorem in contrast to some other approaches. We generalize the technique for calculating particle flux based on the pseudoparticle trajectories for steady state solutions and for time-dependent solutions in 3-D. To validate our code, first we show that good agreement exists between solutions obtained by our steady state stochastic method and a traditional finite difference method. Then we show that good agreement also exists for our time-dependent method for an idealized and simplified heliosphere which has a Parker magnetic field and a simple initial condition for two different inner boundary conditions.