Remaining questions about GC approximation

[henry2004y]

1. Why don't we see an improvement in accuracy when we switch to high order numerical interpolations?
2. Under what conditions can GC tracing get better performance compared to full trajectory tracing?
3. How to include relativity into consideration?
4. Are there any other forms of GC equations that worth implementing?

[TCLiuu]

1. Perhaps the form of the field is relatively simple, or the higher-order correction magnitude is too small to be significant; this still requires further analysis.
2. According to Courant condition, full trajectory tracing requires a time step smaller than the cyclotron period, but the characteristic time scale of the guiding center method corresponds to the characteristic scale of the field.
3. I do not know.
4. In fact, most of the guiding center code currently is based on guiding center equations under a toroidal configuration. However, this might not be universally applicable to TestParticle.jl? Other than that, I don't know.

[henry2004y]

This link describes the test particle model developed at the Center for Geospace Storms (CGS): https://cgs.jhuapl.edu/Models/chimp.php

Specifically, they mentioned that

Particle trajectories are calculated using either the full Lorentz equations of motion or a Hamiltonian guiding center formulation based on the ratio of gyroradius to the local magnetic field curvature.

In Sorathia+ 2017, they mentioned that

For the ion species, H+ and O+, we calculate the trajectories using the full Lorentz equations. These equations are solved using a relativistic Boris algorithm [Birdsall and Langdon, 2005, p. 356] with an adaptive time step, Δt = ϵp/F, where p is the particle momentum, F is the Lorentz force and ϵ is a preselected small number. Electrons are evolved using a hybrid method, utilizing the Hamiltonian formulation of the guiding center equations [Ukhorskiy and Sitnov, 2013] when possible and switching to the full Lorentz equations when the local field conditions render the guiding center approximation invalid. The condition for this switch is $\rho_g |\nabla B| / B > 10^{-3}$, where $\rho_g$ is the particle gyroradius and B/|∇B| defines the characteristic length scale of the magnetic field gradient. The guiding center trajectory is transformed to the full Lorentz trajectory using a randomly generated gyrophase angle.