Numerical methods for solution of the stochastic differential equations equivalent to the non-stationary Parker's transport equation
This work provides numerical methods for solving a specific transport equation in cosmic ray physics, but it is incremental as it applies existing SDE solvers to a known equation.
The authors derive numerical schemes for strong-order integration of SDEs corresponding to the non-stationary Parker transport equation, applying Euler-Maruyama, Milstein, and stochastic Runge-Kutta methods. They discuss accuracy trade-offs but provide no concrete numerical results.
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.