Internal

Public APIs

TestParticle.AdaptiveBoris Method

AdaptiveBoris(; dtmin, dtmax, safety=0.1)

Adaptive Boris method with adaptive time stepping based on local gyrofrequency. The time step is determined by dt = safety / |qB/m|, clamped by dtmin and dtmax.

source

TestParticle.AdaptiveHybrid Type

AdaptiveHybrid{T}

Hybrid algorithm that switches between adaptive Boris (FO) and RK45 (GC).

source

TestParticle.LazyTimeInterpolator Type

LazyTimeInterpolator{T, F, L}

A callable struct for handling time-dependent fields with lazy loading and linear time interpolation.

Fields

• times::Vector{T}: Sorted vector of time points.

• loader::L: Function i -> field that loads the field at index i.

• buffer::Dict{Int, F}: Cache for loaded fields.

• lock::ReentrantLock: Lock for thread safety.

source

TestParticle.TraceGCProblem Type

TraceGCProblem{uType, tType, isinplace, P, F <: AbstractODEFunction, PF} <: AbstractODEProblem{uType, tType, isinplace}

Problem type for tracing guiding centers.

source

TestParticle.TraceHybridProblem Type

TraceHybridProblem{uType, tType, isinplace, P, F <: AbstractODEFunction, PF}

Problem type for hybrid GC-FO tracing. Initial condition u0 should be the full orbit state (6-element).

source

TestParticle.BiKappa Function

 BiKappa(args...; kw...)

Construct a BiKappa distribution. Forwards to VelocityDistributionFunctions.BiKappa.

 BiKappa(B, u0, ppar, pperp, n, kappa; m=mᵢ)

Construct a BiKappa distribution with magnetic field B, bulk velocity u0, parallel thermal pressure ppar, perpendicular thermal pressure pperp, number density n, and spectral index kappa in SI units. The default particle is proton.

source

TestParticle.BiMaxwellian Function

 BiMaxwellian(args...; kw...)

Construct a BiMaxwellian distribution. Forwards to VelocityDistributionFunctions.BiMaxwellian.

 BiMaxwellian(B, u0, ppar, pperp, n; m=mᵢ)

Construct a BiMaxwellian distribution with magnetic field B, bulk velocity u0, parallel thermal pressure ppar, perpendicular thermal pressure pperp, and number density n in SI units. The default particle is proton.

source

TestParticle.Kappa Function

 Kappa(args...; kw...)

Construct a Kappa distribution. Forwards to VelocityDistributionFunctions.Kappa.

 Kappa(u0, p, n, kappa; m=mᵢ)

Construct a Kappa distribution with bulk velocity u0, thermal pressure p, number density n, and spectral index kappa in SI units. The default particle is proton.

source

TestParticle.Maxwellian Function

 Maxwellian(args...; kw...)

Construct a Maxwellian distribution. Forwards to VelocityDistributionFunctions.Maxwellian.

 Maxwellian(u0, p, n; m=mᵢ)

Construct a Maxwellian distribution with bulk velocity u0, thermal pressure p, and number density n in SI units. The default particle is proton.

source

TestParticle.energy2velocity Method

Return velocity magnitude from energy in [eV].

source

TestParticle.full_to_gc Method

full_to_gc(xu, param)

Convert full particle state xu to guiding center state state_gc and magnetic moment μ. Returns (state_gc, μ), where state_gc = [R..., vpar].

source

TestParticle.gc_to_full Function

gc_to_full(state_gc, param, μ, phase=0)

Convert guiding center state state_gc to full particle state xu. Returns xu = [x, y, z, vx, vy, vz].

source

TestParticle.get_adiabaticity Function

get_adiabaticity(r, Bfunc, q, m, μ, t=0.0)
get_adiabaticity(r, Bfunc, μ, t=0.0; species = Proton)

Calculate the adiabaticity parameter ϵ = ρ / Rc at position r and time t. ρ is the gyroradius and Rc is the radius of curvature of the magnetic field.

source

TestParticle.get_adiabaticity Method

get_adiabaticity(sol::AbstractODESolution, t)

Calculate the adiabaticity parameter ϵ from a solution sol at time t. t must be a scalar.

source

TestParticle.get_adiabaticity Method

get_adiabaticity(sol::AbstractODESolution)

Calculate the adiabaticity parameter ϵ from a solution sol at all time steps. The parameters q, m, Bfunc are extracted from sol.prob.p.

source

TestParticle.get_curvature_radius Method

get_curvature_radius(x, t, Bfunc)

Calculate the radius of curvature of the magnetic field at position x and time t. Returns Inf if the field is zero or the field lines are straight.

source

TestParticle.get_energy Method

Calculate the energy [eV] of a relativistic particle from γv in [m/s].

source

TestParticle.get_energy Method

Return the energy [eV] from relativistic sol.

source

TestParticle.get_fields Method

get_fields(sol::AbstractODESolution)

Return the electric and magnetic fields from the solution sol.

source

TestParticle.get_gc Method

get_gc(xu, param)
get_gc(x, y, z, vx, vy, vz, bx, by, bz, q2m)

Calculate the coordinates of the guiding center according to the phase space coordinates of a particle. Reference: wiki

Nonrelativistic definition:

$$\mathbf{X}=\mathbf{x}-m\frac{\mathbf{b}\times \mathbf{v}}{qB}$$source

TestParticle.get_gc_func Method

get_gc_func(param)

Return the function for plotting the orbit of guiding center.

Example

param = prepare(E, B; species = Proton)
# The definitions of stateinit, tspan, E and B are ignored.
prob = ODEProblem(trace!, stateinit, tspan, param)
sol = solve(prob, Vern7(); dt = 2e-11)

f = Figure(fontsize = 18)
ax = Axis3(f[1, 1], aspect = :data)
gc = param |> get_gc_func
gc_plot(x, y, z, vx, vy, vz) = (gc(SA[x, y, z, vx, vy, vz])...,)
lines!(ax, sol, idxs = (gc_plot, 1, 2, 3, 4, 5, 6))

source

TestParticle.get_gc_velocity Method

get_gc_velocity(y, p, t)

Get the guiding center velocity.

source

TestParticle.get_gyrofrequency Function

get_gyrofrequency(B=5e-9; q=qᵢ, m=mᵢ)

Return the gyrofrequency [rad/s].

Arguments

• B: Magnetic field magnitude [T]. Default is 5 nT.

• q: Charge [C]. Default is proton charge.

• m: Mass [kg]. Default is proton mass.

source

TestParticle.get_gyroperiod Function

get_gyroperiod(B=5e-9; q=qᵢ, m=mᵢ)

Return the gyroperiod [s].

Arguments

• B: Magnetic field magnitude [T]. Default is 5 nT.

• q: Charge [C]. Default is proton charge.

• m: Mass [kg]. Default is proton mass.

source

TestParticle.get_gyroradius Method

get_gyroradius(V, B; q=qᵢ, m=mᵢ)

Return the gyroradius [m].

Arguments

• V: Velocity magnitude [m/s] (usually perpendicular to the magnetic field).

• B: Magnetic field magnitude [T].

• q: Charge [C]. Default is proton charge.

• m: Mass [kg]. Default is proton mass.

source

TestParticle.get_gyroradius Method

get_gyroradius(sol::AbstractODESolution, t)

Return the gyroradius [m] from the solution sol at time t. The interpolated magnetic field function is obtained from sol.prob.p.

source

TestParticle.get_mean_magnitude Method

get_mean_magnitude(B)

Calculate the Root Mean Square (RMS) magnitude of a vector field B. It is assumed that the first dimension of B represents the vector components. This function is compatible with any spatial dimension.

source

TestParticle.get_number_density_flux Method

get_number_density_flux(grid::CartesianGrid, sols, dt)

Calculate the steady state particle number density flux on a uniform Cartesian grid. The flux is estimated by accumulating the number of particles in each cell at time steps dt, divided by the surface area of each cell.

Arguments

• grid: A CartesianGrid from Meshes.jl.

• sols: Particle trajectory solutions (e.g. EnsembleSolution).

• dt: Time step for sampling particle positions.

source

TestParticle.get_velocity Method

Return velocity from relativistic γv in sol.

source

TestParticle.get_work Method

get_work(sol::AbstractODESolution)

Return the work done by the electric field from the solution sol.

source

TestParticle.prepare Function

prepare(args...; kwargs...) -> (q2m, m, E, B, F)
prepare(E, B, F = ZeroField(); kwargs...)
prepare(grid::CartesianGrid, E, B, F = ZeroField(); kwargs...)
prepare(x, E, B, F = ZeroField(); dir = 1, kwargs...)
prepare(x, y, E, B, F = ZeroField(); kwargs...)
prepare(x, y, z, E, B, F = ZeroField(); kwargs...)
prepare(B; E = ZeroField(), F = ZeroField(), kwargs...)

Return a tuple consists of particle charge-mass ratio for a prescribed species of charge q and mass m, mass m for a prescribed species, analytic/interpolated EM field functions, and external force F.

Prescribed species are Electron and Proton; other species can be manually specified with m and q keywords or species = Ion(m̄, q̄), where m̄ and q̄ are the mass and charge numbers respectively.

Direct range input for uniform grid in 1/2/3D is supported. For 1D grid, an additional keyword dir is used for specifying the spatial direction, 1 -> x, 2 -> y, 3 -> z. For 3D grid, the default grid type is CartesianGrid. To use StructuredGrid (spherical) grid, an additional keyword gridtype is needed. For StructuredGrid (spherical) grid, dimensions of field arrays should be (Br, Bθ, Bϕ).

Keywords

• order::Int=1: order of interpolation in [1,2,3].

• bc::Int=1: type of boundary conditions, 1 -> NaN, 2 -> periodic.

• species=Proton: particle species.

• q=nothing: particle charge.

• m=nothing: particle mass.

• gridtype: CartesianGrid, RectilinearGrid, StructuredGrid.

source

TestParticle.prepare_gc Method

prepare_gc(xv, xrange, yrange, zrange, E, B;
    species = Proton, q = nothing, m = nothing, order::Int = 1, bc::Int = 1)
prepare_gc(xv, E, B; species = Proton, q = nothing, m = nothing)

Prepare the guiding center parameters for a particle.

source

TestParticle.sample_unit_sphere Method

sample_unit_sphere()

Sample a unit vector on a sphere uniformly.

source

TestParticle.trace! Method

trace!(dy, y, p, t)

ODE equations for charged particle moving in EM field and external force field with in-place form.

source

TestParticle.trace Method

trace(y, p, t)::SVector{6}

ODE equations for charged particle moving in EM field and external force field with out-of-place form.

source

TestParticle.trace_fieldline! Method

trace_fieldline!(dx, x, p, s)

Equation for tracing magnetic field lines with in-place form. The parameter p is the magnetic field function. Note that the independent variable s represents the arc length.

source

TestParticle.trace_fieldline Method

trace_fieldline(x, p, s)

Equation for tracing magnetic field lines with out-of-place form.

source

TestParticle.trace_fieldline Method

trace_fieldline(u0, p, tspan::Tuple{<:Real, <:Real}; mode::Symbol=:both, kw...)

Helper function to create ODEProblems for tracing magnetic field lines.

Arguments

• u0: Initial position.

• p: Parameter tuple containing the magnetic field. It can also be the magnetic field array (numerical) or function (analytical/interpolator).

• tspan: Time span (arc length) for tracing. typically (0.0, L).

• mode: Tracing mode, one of :forward, :backward, :both.

• kw: Keyword arguments passed to prepare when p is an array.

Returns

• If mode is :forward or :backward, returns a single ODEProblem.

• If mode is :both, returns a vector of two ODEProblems (forward and backward).

source

TestParticle.trace_gc! Method

trace_gc!(dy, y, p, t)

Guiding center equations for nonrelativistic charged particle moving in EM field with in-place form. Variable y = (x, y, z, u), where u is the velocity along the magnetic field at (x,y,z).

source

TestParticle.trace_gc_drifts! Method

trace_gc_drifts!(dx, x, p, t)

Equations for tracing the guiding center using analytical drifts, including the grad-B drift, curvature drift, and ExB drift. Parallel velocity is also added. This expression requires the full particle trajectory p.sol.

source

TestParticle.trace_gc_exb! Method

trace_gc_exb!(dx, x, p, t)

Equations for tracing the guiding center using the ExB drift and parallel velocity from a reference trajectory.

source

TestParticle.trace_gc_flr! Method

trace_gc_flr!(dx, x, p, t)

Equations for tracing the guiding center using the ExB drift with FLR corrections and parallel velocity.

source

TestParticle.trace_normalized! Method

trace_normalized!(dy, y, p, t)

Normalized ODE equations for charged particle moving in EM field with in-place form. If the field is in 2D X-Y plane, periodic boundary should be applied for the field in z via the extrapolation function provided by Interpolations.jl.

source

TestParticle.trace_normalized Method

trace_normalized(y, p, t)

Normalized ODE equations for charged particle moving in EM field with out-of-place form.

source

TestParticle.trace_relativistic! Method

trace_relativistic!(dy, y, p, t)

ODE equations for relativistic charged particle (x, γv) moving in EM field with in-place form.

source

TestParticle.trace_relativistic Method

trace_relativistic(y, p, t) -> SVector{6}

ODE equations for relativistic charged particle (x, γv) moving in static EM field with out-of-place form.

source

TestParticle.trace_relativistic_normalized! Method

trace_relativistic_normalized!(dy, y, p, t)

Normalized ODE equations for relativistic charged particle (x, γv) moving in EM field with in-place form.

source

TestParticle.trace_relativistic_normalized Method

trace_relativistic_normalized(y, p, t)

Normalized ODE equations for relativistic charged particle (x, γv) moving in EM field with out-of-place form.

source

Private types and methods

TestParticle.AbstractFieldInterpolator Type

AbstractFieldInterpolator

Abstract type for all field interpolators.

source

TestParticle.AbstractTraceSolution Type

Abstract type for tracing solutions.

source

TestParticle.Field Type

Field{itd, F} <: AbstractField{itd}

A representation of a field function f, defined by:

time-independent field

$$\mathbf{F}=F(\mathbf{x}),$$

time-dependent field

$$\mathbf{F}=F(\mathbf{x},t).$$

Arguments

• field_function::Function: the function of field.

• itd::Bool: whether the field function is time dependent.

• F: the type of field_function.

source

TestParticle.FieldInterpolator Type

FieldInterpolator{T}

A callable struct that wraps a 3D interpolation object.

source

TestParticle.FieldInterpolator1D Type

FieldInterpolator1D{T}

A callable struct that wraps a 1D interpolation object.

source

TestParticle.FieldInterpolator2D Type

FieldInterpolator2D{T}

A callable struct that wraps a 2D interpolation object.

source

TestParticle.Species Type

Type for the particles: Proton, Electron.

source

TestParticle.SphericalFieldInterpolator Type

SphericalFieldInterpolator{T}

A callable struct for spherical grid interpolation (scalar or combined vector).

source

TestParticle.VDF Type

Abstract type for velocity distribution functions.

source

TestParticle._boris! Method

Apply Boris method for particles with index in irange.

source

TestParticle._boris_loop! Method

Apply Boris method for particles with index in irange.

source

TestParticle._prepare Method

Prepare for advancing.

source

TestParticle._rk4! Method

Apply RK4 method for particles with index in irange.

source

TestParticle._rk45! Method

Apply RK45 method for particles with index in irange.

source

TestParticle._solve_single_adaptive_boris Method

See _solve_single_boris for the rationale behind the single_prob construction.

source

TestParticle._solve_single_boris Method

_solve_single_boris(prob, i, ...)

Solve a single trajectory i of prob for use with EnsembleDistributed.

_generic_boris! uses the loop index i for two purposes simultaneously: applying prob_func(prob, i, false) to select per-particle initial conditions, and storing the result at sols[i]. For a distributed worker handling only one trajectory at a time, a 1-element local_sols would be out of bounds if i > 1 were passed directly. Decoupling the two uses would require threading a storage offset through the entire _dispatch_boris! → _generic_boris! call chain.

Instead, we pre-apply prob_func to get the correct IC for trajectory i and wrap the result in a fresh TraceProblem with the default (identity) prob_func, so _generic_boris! can safely iterate 1:1 without applying prob_func again. The TraceProblem construction is a negligible struct copy relative to the simulation cost and the serialization overhead inherent in pmap.

source

TestParticle._strip_units Method

Helper function to strip units from values.

source

TestParticle.adapt_field_to_gpu Method

adapt_field_to_gpu(field::Field, backend::KA.Backend)

Adapt interpolation fields to GPU memory using Adapt.jl. Analytic functions are returned unchanged.

source

TestParticle.boris_velocity_update Method

boris_velocity_update(v, E, B, qdt_2m)

Update velocity using the Boris method, returning the new velocity as an SVector. This is the core logic shared between the standard solver and the kernel solver.

source

TestParticle.cart2sph Method

cart2sph(x, y, z)

Convert from Cartesian to spherical coordinates vector.

source

TestParticle.cross! Method

In-place cross product.

source

TestParticle.get_cell_centers Method

Return cell center coordinates from 2D/3D CartesianGrid.

source

TestParticle.get_cell_centers Method

Return cell center coordinates from 2D/3D RectilinearGrid.

source

TestParticle.get_curvature Method

get_curvature(x, t, Bfunc)

Calculate the curvature vector κ of the magnetic field at position x and time t.

source

TestParticle.get_interpolator Method

get_interpolator(A, gridx, gridy, gridz, order::Int=1, bc::Int=1)
get_interpolator(gridtype, A, grid1, grid2, grid3, order::Int=1, bc::Int=1)

Return a function for interpolating field array A on the grid given by gridx, gridy, and gridz.

Arguments

• gridtype: CartesianGrid, RectilinearGrid or StructuredGrid.

• A: field array. For vector field, the first dimension should be 3.

• order::Int=1: order of interpolation in [1,2,3].

• bc::Int=1: type of boundary conditions, 1 -> NaN, 2 -> periodic, 3 -> Flat.

Notes

The input array A may be modified in-place for memory optimization.

source

TestParticle.get_magnetic_properties Method

get_magnetic_properties(x, t, Bfunc)

Calculate magnetic field properties at position x and time t. Returns tuple (B, ∇B, κ, b̂, Bmag):

• B: Magnetic field vector

• ∇B: Gradient of magnetic field magnitude

• κ: Curvature vector

• b̂: Unit magnetic field vector

• Bmag: Magnitude of B

source

TestParticle.get_number_density Method

get_number_density(sols, grid, t_start, t_end, dt)

Calculate time-averaged particle number density from t_start to t_end with step dt.

source

TestParticle.get_number_density Method

get_number_density(sols, grid, t)

Calculate particle number density at time t in a given grid.

source

TestParticle.get_perp_vector Method

get_perp_vector(b::AbstractVector)

Obtain two unit vectors e1 and e2 such that (e1, e2, b) form a right-handed orthonormal system.

source

TestParticle.get_rotation_matrix Method

get_rotation_matrix(axis::AbstractVector, angle) :: SMatrix{3,3}

Create a rotation matrix for rotating a 3D vector around a unit axis by an angle in radians. Reference: Rotation matrix from axis and angle

Example

using LinearAlgebra
v = [-0.5, 1.0, 1.0]
v̂ = normalize(v)
θ = deg2rad(-74)
R = get_rotation_matrix(v̂, θ)

source

TestParticle.get_work_rates Function

get_work_rates(xu, p, t)

Calculate the work rates done by the electric field and the betatron acceleration. Returns a tuple (P_par, P_fermi, P_grad, P_betatron).

source

TestParticle.get_work_rates_gc Method

get_work_rates_gc(xv, p, t)

Calculate the work rates done by the electric field and the betatron acceleration for guiding center.

source

TestParticle.getinterp Function

getinterp(::Type{<:CartesianGrid}, A, gridx, gridy, gridz, order::Int=1, bc::Int=1)

Return a function for interpolating field array A on the grid given by gridx, gridy, and gridz.

Arguments

• order::Int=1: order of interpolation in [1,2,3].

• bc::Int=1: type of boundary conditions, 1 -> NaN, 2 -> periodic, 3 -> Flat.

• dir::Int: 1/2/3, representing x/y/z direction.

Notes

The input array A may be modified in-place for memory optimization.

source

TestParticle.getinterp_scalar Function

getinterp_scalar(::Type{<:CartesianGrid}, A, gridx, gridy, gridz, order::Int=1, bc::Int=1)

Return a function for interpolating scalar array A on the grid given by gridx, gridy, and gridz. Currently only 3D arrays are supported.

Arguments

• order::Int=1: order of interpolation in [1,2,3].

• bc::Int=1: type of boundary conditions, 1 -> NaN, 2 -> periodic, 3 -> Flat.

• dir::Int: 1/2/3, representing x/y/z direction.

source

TestParticle.is_time_dependent Method

Judge whether the field function is time dependent.

source

TestParticle.makegrid Method

Return uniform range from 2D/3D CartesianGrid.

source

TestParticle.makegrid Method

Return ranges from 2D/3D Meshes.jl RectilinearGrid.

source

TestParticle.solve Function

solve(prob::TraceGCProblem; trajectories::Int=1, dt::AbstractFloat,
savestepinterval::Int=1, isoutofdomain::Function=ODE_DEFAULT_ISOUTOFDOMAIN,
alg::Symbol=:rk4, abstol=1e-6, reltol=1e-6, maxiters=10000,
save_fields::Bool=false, save_work::Bool=false)

Trace guiding centers using the RK4 method with specified prob. If alg is :rk45, uses adaptive time stepping.

source

TestParticle.solve Method

solve(prob::TraceProblem; trajectories::Int=1, dt::AbstractFloat,
    savestepinterval::Int=1, isoutofdomain::Function=ODE_DEFAULT_ISOUTOFDOMAIN,
    n::Int=1, save_start::Bool=true, save_end::Bool=true, save_everystep::Bool=true,
    save_fields::Bool=false, save_work::Bool=false)

Trace particles using the Boris method with specified prob.

keywords

• trajectories::Int: number of trajectories to trace.

• dt::AbstractFloat: time step.

• savestepinterval::Int: saving output interval.

• isoutofdomain::Function: a function with input of position and velocity vector xv that determines whether to stop tracing.

• n::Int: number of substeps for the Multistep Boris method. Default is 1 (standard Boris).

• save_start::Bool: save the initial condition. Default is true.

• save_end::Bool: save the final condition. Default is true.

• save_everystep::Bool: save the state at every savestepinterval. Default is true.

• save_fields::Bool: save the electric and magnetic fields. Default is false.

• save_work::Bool: save the work done by the electric field. Default is false.

source

TestParticle.sph2cart Method

sph2cart(r, θ, ϕ)

Convert from spherical to Cartesian coordinates vector.

source

TestParticle.sph_to_cart_vector Method

sph_to_cart_vector(vr, vθ, vϕ, θ, ϕ)

Convert a vector from spherical to Cartesian coordinates. θ and ϕ are defined in radians.

source

TestParticle.trace_gc Method

trace_gc(y, p, t)

Guiding center equations for nonrelativistic charged particle moving in EM field with out-of-place form.

source

TestParticle.update_dp5 Method

update_dp5(y, param, dt, t)::(dy, E)

Update state using the Dormand-Prince 5(4) method. Returns dy as the update and E as the error estimate, both as SArrays.

source

TestParticle.update_location! Method

Update location in one timestep dt.

source

TestParticle.update_rk4 Method

update_rk4(y, param, dt, t)

Update state using the RK4 method. Returns dy as the update SArray.

source

TestParticle.update_velocity! Method

update_velocity!(xv, paramBoris, param, dt, t)

Update velocity using the Boris method, Birdsall, Plasma Physics via Computer Simulation. Reference: DTIC

source

TestParticle.update_velocity Method

update_velocity(v, r, param, dt, t)

Update velocity using the Boris method, returning the new velocity as an SVector.

source

TestParticle.update_velocity_multistep Method

update_velocity_multistep(v, r, param, dt, t, n)

Update velocity using the Multistep Boris method, returning the new velocity as an SVector. Reference: Zenitani & Kato 2025

source