BorisIntegrator

class plasmapy.simulation.particle_integrators.BorisIntegrator[source]

Bases: AbstractIntegrator

The explicit Boris pusher.

Attributes Summary

is_relativistic

The explicit Boris pusher is not relativistic.

Methods Summary

push(x, v, B, E, q, m, dt)

param x:

Particle position at full timestep, in SI (meter) units.

Attributes Documentation

is_relativistic

The explicit Boris pusher is not relativistic.

Methods Documentation

static push(x, v, B, E, q, m, dt)[source]
Parameters:

  • x (ndarray) – Particle position at full timestep, in SI (meter) units.

  • v (ndarray) – Particle velocity at half timestep, in SI (meter/second) units.

  • B (ndarray) – Magnetic field at full timestep, in SI (tesla) units.

  • E (float) – Electric field at full timestep, in SI (V/m) units.

  • q (float) – Particle charge, in SI (coulomb) units.

  • m (float) – Particle mass, in SI (kg) units.

  • dt (float) – Timestep, in SI (second) units.

Returns:

  • x (ndarray) – Particle position x after Boris push algorithm, in SI (meter) units.

  • v (ndarray) – Particle velocity after Boris push algorithm, in SI (meter/second) units.

Examples

>>> B = np.array([[0.0, 0.0, 5.0]])
>>> E = np.array([[0.0, 0.0, 0.0]])
>>> x_t0 = np.array([[0.0, 0.0, 0.0]])
>>> v_t0 = np.array([[5.0, 0.0, 0.0]])
>>> x_t1, v_t1 = BorisIntegrator.push(
...     x=x_t0, v=v_t0, B=B, E=E, q=1.0, m=1.0, dt=0.01
... )
>>> x_t1
array([[ 0.04993754, -0.00249844,  0.        ]])
>>> v_t1
array([[ 4.9937539 , -0.24984385,  0.        ]])
>>> x_t1, v_t1 = BorisIntegrator.push(
...     x=x_t0, v=v_t0, B=B, E=E, q=1.0, m=1.0, dt=0.01
... )
>>> x_t1
array([[ 0.04993754, -0.00249844,  0.        ]])
>>> v_t1
array([[ 4.9937539 , -0.24984385,  0.        ]])
>>> # B parallel to v
>>> B = np.array([[0.0, 0.0, 5.0]])
>>> v_t0 = np.array([[0.0, 0.0, 5.0]])
>>> x_t0 = np.array([[0.0, 0.0, 0.0]])
>>> x_t1, v_t1 = BorisIntegrator.push(
...     x=x_t0, v=v_t0, B=B, E=E, q=1.0, m=1.0, dt=0.01
... )
>>> # no rotation of vector v
>>> v_t1
array([[0., 0., 5.]])
>>> x_t1
array([[0.  , 0.  , 0.05]])
>>> # B perpendicular to v
>>> B = np.array([[5.0, 0.0, 0.0]])
>>> v_t0 = np.array([[0.0, 5.0, 0.0]])
>>> x_t0 = np.array([[0.0, 0.0, 0.0]])
>>> x_t1, v_t1 = BorisIntegrator.push(
...     x=x_t0, v=v_t0, B=B, E=E, q=1.0, m=1.0, dt=0.01
... )
>>> # rotation of vector v
>>> v_t1
array([[ 0.        ,  4.9937539 , -0.24984385]])
>>> x_t1
array([[ 0.        ,  0.04993754, -0.00249844]])
>>> # nonzero E and zero B
>>> E = np.array([[1.0, 1.0, 1.0]])
>>> B = np.array([[0.0, 0.0, 0.0]])
>>> v_t0 = np.array([[0.0, 0.0, 5.0]])
>>> x_t0 = np.array([[0.0, 0.0, 0.0]])
>>> x_t1, v_t1 = BorisIntegrator.push(
...     x=x_t0, v=v_t0, B=B, E=E, q=1.0, m=1.0, dt=0.01
... )
>>> v_t1
array([[0.01, 0.01, 5.01]])
>>> x_t1
array([[0.0001, 0.0001, 0.0501]])
>>> x_t1, v_t1 = BorisIntegrator.push(
...     x=x_t0, v=v_t0, B=B, E=E, q=1.0, m=1.0, dt=0.01
... )
>>> v_t1
array([[0.02, 0.02, 5.02]])
>>> x_t1
array([[0.0001, 0.0001, 0.0501]])

Notes

The Boris algorithm [Boris, 1970] is the standard energy conserving algorithm for particle movement in plasma physics. See Birdsall and Langdon [2004] for more details, and this page on the Boris method for a nice overview.

Conceptually, the algorithm has three phases:

  1. Add half the impulse from electric field.

  2. Rotate the particle velocity about the direction of the magnetic field.

  3. Add the second half of the impulse from the electric field.

This ends up causing the magnetic field action to be properly “centered” in time, and the algorithm, being a symplectic integrator, conserves energy.