The Boris Particle Pusher
The Boris algorithm advances a charged particle in electromagnetic fields by splitting the motion into a half electric kick, a magnetic rotation, and a second kick.
A stable rotation for magnetized particles
Charged particles in a magnetic field gyrate, and a naive integrator either spirals in or out over many orbits, corrupting long simulations. The Boris method solves this by splitting the Lorentz-force update into three exact substeps within a leapfrog scheme: a half acceleration by the electric field, a pure rotation by the magnetic field, and a second half electric acceleration.
The key insight is that the magnetic force does no work and only rotates the velocity. The Boris rotation reproduces this property discretely, so the particle's speed is preserved through the magnetic substep and the gyration radius stays bounded over arbitrarily many orbits.
Why it endures
The Boris pusher is time-reversible and phase-space volume preserving in the magnetic rotation, giving excellent long-term energy behavior without being formally symplectic. It is second-order accurate, requires only one field evaluation per step, and is cheap enough to apply to billions of particles. These properties explain its near-universal use in particle-in-cell codes.
The rotation in practice
The magnetic rotation is implemented with two auxiliary vectors t and s built from the field and timestep. The velocity is first rotated by an intermediate step using t, then reflected using s, reproducing the exact finite-angle rotation without trigonometric functions.
- Leapfrog structure: velocity and position staggered in time
- Exact energy conservation through the magnetic substep
- No secular drift in gyroradius over many orbits
- Trigonometry-free implementation using auxiliary vectors
Because faithful long-time orbit integration is the foundation of kinetic plasma modeling, the Boris pusher is a standard component wherever magnetized particle motion is simulated for confinement or direct-conversion studies.