Graduate Velocity Verlet for relativistic simulation

[Philip Koeck]

I'm simulating a situation that's partly relativistic and I'm wondering if it's wise to use Velocity Verlet.

A fast electron (200 keV or roughly 208 000 000 m/s) travels along the z-axis and intersects a beam of slower electrons (1 keV or roughly 20 000 000 m/s) that are moving along the x-axis.

I treat the slower electrons as non-relativistic.

For the fast electron I assume it's travelling essentially in the z-direction at all times, which is very accurate as far as I can see.
So I use the parallel corrected mass in the z-direction and the orthogonal in x and y to get the acceleration of the fast electron at every time step.

I realise that velocity verlet is not intended for accelerations that are velocity-dependent, but in this case the velocity of the fast electron is almost constant so γ and γ³ are almost constant during the whole simulation.

Does it sound okay to use Velocity Verlet in this case or should I consider a different algorithm?

 

[Ibix]

I don't know the answer to your question about the integrator, but you could try working with four-vectors. The four velocity has a constant magnitude by definition and the Lorentz force law relates to it via the four momentum which is the invariant mass times the four velocity.

 

[Philip Koeck]

This sounds quite promising: https://www.unige.ch/~hairer/preprints/relcpd.pdf (https://epubs.siam.org/doi/10.1137/23M1568946).
In section 4 they discuss an energy preserving algorithm.

Thanks for pointing me in that direction.