Particle Pushing

We use Boris Method [Bor70]. This method is well described in [BL85] (ch. 4-3) (or more recent reprint) which is a standrart in Particl-in-Cell technique for its good numerical properties:

Pros

  • Second order accurate
  • Error is not cummulative (as compared to Runge-Kutta)
  • Fields are evelautaed only once

Cons

  • The method requies to process particle array twice: This is required when used in hybrid codes as we perform only half time-step advance.

Derivation

We are solving equation:

\[\frac{\mathbf{v}^{n+1/2} - \mathbf{v}^{n-1/2}}{\triangle t} = \frac{q}{m} \left[ \mathbf{E}^n + \mathbf{v}^{n} \times \mathbf{B}^n\right]\]

where \(\mathbf{v}^{n}\) can be expressed as an average of two times

\[\mathbf{v}^{n} = \frac{1}{2}(\mathbf{v}^{n+1/2} + \mathbf{v}^{n-1/2}).\]

Lets perform following substitutions:

\[\mathbf{v}^{n+1/2} = \mathbf{v}^{+} + \frac{q \mathbf{E}^n}{m} \frac{\triangle t}{2}\]\[\mathbf{v}^{n-1/2} = \mathbf{v}^{-} - \frac{q \mathbf{E}^n}{m} \frac{\triangle t}{2}\]

by doing so, we split the particle push into rotation around magnetic field \(\mathbf{B}^n\) and acceleration by electric field \(\mathbf{E}^n\). The rotation step is governed by the equation:

\[\mathbf{v}^{+} - \mathbf{v}^{-} = \frac{\triangle t}{2} \frac{q}{m} (\mathbf{v}^{+} + \mathbf{v}^{-}) \times \mathbf{B}\]

Steps

  1. \(\mathbf{v}^- = \mathbf{v}^{n-1/2} + \frac{q \mathbf{E}^n}{m} \frac{\triangle t}{2}\)
  2. \(\mathbf{t} = \mathbf{B}^n \frac{q}{m} \frac{\triangle t}{2}\)
  3. \(\mathbf{s} = 2 \mathbf{t}/ (1 + t^2)\) (from the requirement \(|\mathbf{v}^-|^2=|\mathbf{v}^+|^2\))
  4. \(\mathbf{v}^\prime = \mathbf{v}^- + \mathbf{v}^- \times \mathbf{t}\)
  5. \(\mathbf{v}^+ = \mathbf{v}^- + \mathbf{v}^\prime \times \mathbf{s}\)
  6. \(\mathbf{v}^{n + 1/2} = \mathbf{v}^+ + \frac{q \mathbf{E}^n}{m} \frac{\triangle t}{2}\)

References

[BL85]C K Birdsall and A Bruce Langdon. Plasma Physics via Computer simulations. Graw-Hill, New York, 1985.
[Bor70]Jay P Boris. Acceleration calculation from a scalar potential. Technical Report, Princeton Univ., NJ Plasma Physics Lab., 1970.