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.

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TestParticle.CurrentLoop Type

CurrentLoop{T}

A circular current loop.

Fields

• radius::T: Radius of the loop [m].

• current::T: Current in the loop [A].

• center::SVector{3, T}: Position of the loop center [m].

• normal::SVector{3, T}: Unit normal vector of the loop.

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TestParticle.BiKappa Method

 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.

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TestParticle.BiMaxwellian Method

 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.

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TestParticle.Kappa Method

 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.

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TestParticle.Maxwellian Method

 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.

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TestParticle.energy2velocity Method

Return velocity magnitude from energy in [eV].

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TestParticle.getB_bottle Method

 getB_bottle(x, y, z, distance, a, b, I1, I2) :: StaticVector{3}

Get magnetic field from a magnetic bottle. Reference: wiki

Arguments

• x,y,z::Float: particle coordinates in [m].

• distance::Float: distance between solenoids in [m].

• a::Float: radius of each side coil in [m].

• b::Float: radius of central coil in [m].

• I1::Float: current in the solenoid times number of windings in side coils in [A].

• I2::Float: current in the central solenoid times number of windings in the central loop in [A].

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TestParticle.getB_loop Method

getB_loop(r, R, a, I, n)

Calculate the magnetic field B [T] at point r from a current loop with current I [A], radius a [m], centered at R, and normal vector n.

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TestParticle.getB_loop Method

getB_loop(r, loop::CurrentLoop)

Calculate the magnetic field B [T] at point r from a CurrentLoop.

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TestParticle.getB_mirror Method

 getB_mirror(x, y, z, distance, a, I1) :: StaticVector{3}

Get magnetic field at [x, y, z] from a magnetic mirror generated from two coils.

Arguments

• x,y,z: particle coordinates in [m].

• distance: distance between solenoids in [m].

• a: radius of each side coil in [m].

• I1: current in the solenoid times number of windings in side coils.

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TestParticle.getB_tokamak_coil Method

 getB_tokamak_coil(x, y, z, a, b, ICoils, IPlasma) -> StaticVector{3}

Get the magnetic field from a Tokamak topology consists of 16 coils. Original: Tokamak-Fusion-Reactor

Arguments

• x,y,z: location in [m].

• a: radius of each coil in [m].

• b: radius of central region in [m].

• ICoil: current in the coil times number of windings in [A].

• IPlasma: current of the plasma in [A].

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TestParticle.getB_zpinch Method

 getB_zpinch(x, y, z, I, a) -> StaticVector{Float64, 3}

Get magnetic field from a Z-pinch configuration. Reference: Z-pinch

Arguments

• x,y,z::Float: particle coordinates in [m].

• I::Float: current in the wire [A].

• a::Float: radius of the wire [m].

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TestParticle.get_energy Method

Calculate the energy [eV] of a relativistic particle from γv.

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TestParticle.get_energy Method

Return the energy [eV] from relativistic sol.

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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))

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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.

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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.

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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.

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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.

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TestParticle.get_velocity Method

Return velocity from relativistic γv in sol.

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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.

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TestParticle.sample_unit_sphere Method

sample_unit_sphere()

Sample a unit vector on a sphere uniformly.

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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.

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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.

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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.

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TestParticle.trace_fieldline Method

trace_fieldline(x, p, s)

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

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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).

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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).

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TestParticle.trace_gc_1st! Method

1st order approximation of guiding center equations.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Private types and methods

TestParticle.AbstractTraceSolution Type

Abstract type for tracing solutions.

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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.

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TestParticle.GCTuple Type

The type of parameter tuple for guiding center problem.

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TestParticle.Species Type

Type for the particles: Proton, Electron.

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TestParticle.VDF Type

Abstract type for velocity distribution functions.

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TestParticle._boris! Method

Apply Boris method for particles with index in irange.

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TestParticle._prepare Method

Prepare for advancing.

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TestParticle.cart2sph Method

Convert from Cartesian to spherical coordinates vector.

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TestParticle.cross! Method

In-place cross product.

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TestParticle.dipole Method

Calculates the magnetic field from a dipole with magnetic moment M at r.

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TestParticle.dipole_fieldline Function

 dipole_fieldline(ϕ, L=2.5, nP=100)

Creates nP points on one field line of the magnetic field from a dipole. In a centered dipole magnetic field model, the path along a given L shell can be described as r = L*cos²λ, where r is the radial distance (in planetary radii) to a point on the line, λ is its co-latitude, and L is the L-shell of interest.

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TestParticle.getB_CS_harris Function

 getB_CS_harris(B₀, L)

Return the magnetic field at location r near a current sheet with magnetic strength B₀ and sheet length L. The current sheet is assumed to lie in the z = 0 plane.

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TestParticle.getB_dipole Method

Analytic magnetic field function for testing. Return in SI unit.

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TestParticle.getB_tokamak_profile Method

 getB_tokamak_profile(x, y, z, q_profile, a, R₀, Bζ0) :: StaticVector{3}

Reconstruct the magnetic field distribution from a safe factor(q) profile. Reference: Tokamak, 4th Edition, John Wesson.

Arguments

• x,y,z: location in [m].

• q_profile::Function: profile of q. The variable of this function must be the normalized radius.

• a: minor radius [m].

• R₀: major radius [m].

• Bζ0: toroidal magnetic field on axis [T].

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TestParticle.getE_dipole Method

Analytic electric field function for testing.

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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.

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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.

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TestParticle.get_number_density Method

get_number_density(sols, grid, t)

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

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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)

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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.

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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.

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TestParticle.is_time_dependent Method

Judge whether the field function is time dependent.

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TestParticle.makegrid Method

Return uniform range from 2D/3D CartesianGrid.

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TestParticle.makegrid Method

Return ranges from 2D/3D RectilinearGrid.

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TestParticle.set_axes_equal Method

 set_axes_equal(ax)

Set 3D plot axes to equal scale for Matplotlib. Make axes of 3D plot have equal scale so that spheres appear as spheres and cubes as cubes. Required since ax.axis('equal') and ax.set_aspect('equal') don't work on 3D.

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TestParticle.solve Function

solve(prob::TraceProblem; trajectories::Int=1, dt::AbstractFloat,
savestepinterval::Int=1, isoutofdomain::Function=ODE_DEFAULT_ISOUTOFDOMAIN,
    n::Int=1)

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.

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TestParticle.sph2cart Method

Convert from spherical to Cartesian coordinates vector.

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TestParticle.sph_to_cart_vector Method

Convert a vector from spherical to Cartesian.

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TestParticle.update_location! Method

Update location in one timestep dt.

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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

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TestParticle.update_velocity_multistep! Method

update_velocity_multistep!(xv, paramBoris, param, dt, t, n)

Update velocity using the Multistep Boris method. Reference: Zenitani & Kato 2025

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