Boris particle update

BEACH advances one particle step in two stages: it constructs a candidate trajectory, then resolves the first collision or boundary crossing on that trajectory. The candidate combines a predicted-midpoint field sample, a Boris velocity update, and a trapezoidal position update. Public state and checkpoints keep $(\mathbf{x},\mathbf{v})$ at the same time level before and after the step.

State	Time level
Input	$\mathbf{x}^n,\mathbf{v}^n$
Field sample	Predicted position $\mathbf{x}_\mathrm{mid}$
Output candidate	$\mathbf{x}^{n+1},\mathbf{v}^{n+1}$
Magnetic field	Uniform sim.b0

Sample the field at a predicted midpoint

For step interval $\Delta t$, current velocity predicts the midpoint

$$\mathbf{x}_\mathrm{mid}= \mathbf{x}^n+\frac{1}{2}\mathbf{v}^n\Delta t.$$

The field snapshot is evaluated once there:

$$\mathbf{E}_\mathrm{mid}= \mathbf{E}_\mathrm{snapshot}(\mathbf{x}_\mathrm{mid}).$$

The snapshot composes the element-charge field, sim.e0, and any selected periodic zero mode or outer profile exactly once. Particles in one batch share the snapshot formed from element charge at the start of that batch. See Field evaluation for its construction.

With use_box=true, only the field-sample position is mapped into the solver’s valid region.

Axis	Treatment of $\mathbf{x}_\mathrm{mid}$
Both low and high faces periodic	Wrap into the primary box with modulo
Otherwise	Clamp between box_min and box_max

Candidate trajectory endpoints remain in physical coordinates so that the next stage can order a triangle collision against a box-boundary crossing.

Advance velocity with Boris

For particle charge $q$, mass $m$, and $\mathbf{B}=\texttt{sim.b0}$, first apply a half electric acceleration:

$$\mathbf{v}^-= \mathbf{v}^n+\frac{q}{m}\mathbf{E}_\mathrm{mid}\frac{\Delta t}{2}.$$

The magnetic rotation is

$$\mathbf{t}=\frac{q}{m}\mathbf{B}\frac{\Delta t}{2}, \qquad \mathbf{s}=\frac{2\mathbf{t}}{1+\lVert\mathbf{t}\rVert^2},$$ $$\mathbf{v}'=\mathbf{v}^-+\mathbf{v}^-\times\mathbf{t}, \qquad \mathbf{v}^+=\mathbf{v}^-+\mathbf{v}'\times\mathbf{s},$$

followed by the other half electric acceleration:

$$\mathbf{v}^{n+1}= \mathbf{v}^++\frac{q}{m}\mathbf{E}_\mathrm{mid}\frac{\Delta t}{2}.$$

The implementation uses the standard Boris rotation with $t^2=\mathbf{t}\cdot\mathbf{t}$.

Form the candidate position with the trapezoidal rule

After velocity is updated, the candidate position uses same-time input and output velocities in the trapezoidal rule:

$$\mathbf{x}^{n+1}= \mathbf{x}^n+\frac{1}{2} \left(\mathbf{v}^n+\mathbf{v}^{n+1}\right)\Delta t.$$

For a uniform electric field alone, this matches the constant-acceleration velocity and displacement. Regression tests for a smooth spatial electrostatic field verify second-order convergence of candidate position and velocity when combined with the predicted-midpoint field sample.

Split the step at collisions and boundary crossings

The equations above produce a full-step candidate $(\mathbf{x}^{n+1},\mathbf{v}^{n+1})$. It becomes the accepted state when the candidate trajectory has no triangle collision or box-boundary crossing.

Triangle collisions and ordinary box crossings are ordered along the straight trajectory segment within the step that connects $\mathbf{x}^n$ and $\mathbf{x}^{n+1}$. A triangle collision absorbs the particle at the hit point. If a reflection action (reflect or redistributed_reflect) or periodic face is crossed first, BEACH obtains the position from the chord parameter. The event velocity follows the chord tangent and is normalized to the speed implied by the discrete work of the predicted-midpoint electric field,

$$\lVert\mathbf{v}_\mathrm{event}\rVert^2= \lVert\mathbf{v}^n\rVert^2+ 2\frac{q}{m}\mathbf{E}_\mathrm{mid}\cdot (\mathbf{x}_\mathrm{event}-\mathbf{x}^n).$$

BEACH then constructs a new Boris candidate for the remaining interval. This keeps the normal velocity direction consistent with the chord crossing and preserves event speed for any pure magnetic field. A non-positive or non-finite reconstructed speed fails closed. However, BEACH also uses the chord crossing fraction as the remaining-time fraction; it does not solve the exact physical crossing time under acceleration or magnetic rotation. This is a time-discretization approximation of the trajectory-chord event model, so runs that use boundary actions or the potential-barrier decision must check convergence with dt, dt/2, and, when needed, dt/4. At the z-high outer interface, the applicable coupling path refines crossing time from a quadratic trajectory consistent with the Boris endpoints. See Particle collision and boundary events.

Properties of the Boris update

Condition	Property
Pure B	Preserves $\lVert\mathbf{v}\rVert$ to roundoff
Prescribed uniform E/B	Velocity update is reversible under a sign change of $\Delta t$
Smooth prescribed field	Current same-time update is second order in position and velocity
Mesh collision or open-boundary escape	Irreversible process that ends tracking at the collision or boundary point
Self-consistent field changing between batches	Snapshot fixed within a batch and refreshed after charge commit

Choose dt from trajectories and charging results

dt must resolve particle rotation, spatial field variation, and collision geometry:

• cyclotron angle $|q|\lVert\mathbf{B}\rVert\Delta t/m$
• transit time over the spatial electric-field variation scale
• transit time near triangles and box faces
• transit time through a rapidly changing outer-interface profile

Triangle intersections use the straight trajectory segment within a step. When an orbit bends strongly in one step or passes fine geometry, reduce the time step until collision positions converge. Compare dt, dt/2, and when needed dt/4 using trajectories, hit elements, absorption and escape statistics, and post-batch element charge. Keeping max_step fixed halves the maximum physical tracking time when dt is halved, so adjust max_step when comparing the same time horizon.

Code reference

• Boris velocity rotation and trapezoidal position: bem_pusher.f90
• Predicted-midpoint field and remainder reintegration: bem_particle_stepper.f90
• E/B, pure-B conservation, reversibility, and convergence tests: test_dynamics_basic.f90
• Charged-mesh and field-sample tests: test_particle_stepper.f90