Explicit volume-preserving numerical schemes for relativistic trajectories and spin dynamics

A class of explicit numerical schemes is developed to solve for the relativistic dynamics and spin of particles in electromagnetic fields, using the Lorentz–Bargmann-Michel-Telegdi equation formulated in the Clifford algebra representation of Baylis. It is demonstrated that these numerical methods, reminiscent of the leapfrog and Verlet methods, share a number of important properties: they are energy conserving, volume conserving, and second-order convergent. These properties are analyzed empirically by benchmarking against known analytical solutions in constant uniform electrodynamic fields. It is demonstrated that the numerical error in a constant magnetic field remains bounded for long-time simulations in contrast to the Boris pusher, whose angular error increases linearly with time. Finally, the intricate spin dynamics of a particle is investigated in a plane-wave field configuration.

Figure 1

Position $x$ and numerical error $\epsilon \left(t\right)=\parallel \mathit{x}\left(t\right)-{\mathit{x}}_{\mathrm{analytical}}{\left(t\right)\parallel }_{\infty }$ as function of time for the Boris, Verlet-type, and leapfrog-like methods for a particle in a constant electric field. All position curves are overlapping with the analytical solution. The position, the numerical error, and time are expressed in natural units.

Figure 2

Numerical error at $t=t_{\mathrm{final}}=10$ for a particle in a constant electric field as function of the time step $\mathrm{\Delta }t$ for the Boris, Verlet-type, and leapfrog-like methods. The dashed line corresponds to a fit of the data, used to determine the order of convergence. The numerical error and the time step are expressed in natural units.

Figure 3

Relative error on energy as function of time for a particle in a constant magnetic field. Time is expressed in natural units.

Figure 4

Positions $r,\theta$ and numerical errors ${\epsilon }_r,{\epsilon }_{\theta }$ as a function of time for the Boris scheme and a particle in a constant magnetic field. The radius, the numerical error on radius, and time are expressed in natural units.

Figure 5

Positions $r,\theta$ and numerical errors ${\epsilon }_r,{\epsilon }_{\theta }$ as a function of time for the Verlet-type scheme and a particle in a constant magnetic field. The radius, the numerical error on radius, and time are expressed in natural units.

Figure 6

Positions $r,\theta$ and numerical errors ${\epsilon }_r,{\epsilon }_{\theta }$ as a function of time for the leapfrog-like scheme and a particle in a constant magnetic field. The radius, the numerical error on radius, and time are expressed in natural units.

Figure 7

Positions $x,z$ and numerical errors as a function of time for the Boris scheme and a particle in a plane wave. The positions, the numerical error, and time are expressed in natural units.

Figure 8

Positions $x,z$ and numerical errors as a function of time for the Verlet-type scheme and a particle in a plane wave. The positions, the numerical error, and time are expressed in natural units.

Figure 9

Spins $S_x,S_z$ and numerical errors as a function of time for the Verlet-type scheme and a particle in a plane wave. Time is expressed in natural units.