An Overview of Relativistic Particle Pushers and their Extension to Arbitrary Order Accuracy

Abstract

Particle in Cell (PIC) simulations have become a vital tool for the investigation of kinetic processes in plasma physics. Many of the systems investigated with PIC simulations contain particles with relativistic velocities. The correct integration and the knowledge of possible sources of errors in relativistic particle trajectories is of importance to accurately judge the validity of the simulation results. Over the past few decades, various new integration schemes for relativistic particle trajectories in PIC simulations have been proposed. These are aimed at improving numerical accuracy in specific scenarios. This article presents a comprehensive comparison of particle pushers with a focus on explicit schemes. An important class of these schemes is found to be generalisable to arbitrary high order. A comparison of the fourth order variants of these schemes with their second order counterpart is also presented.

Figures (7)

• Figure 1: The $y$--coordinate of the particle trajectory vs time in a crossed electric and magnetic field with $\mathbf{E} = -\mathbf{v}_0\times\mathbf{B}$ (test case B) for a selection of integration schemes. Time is normalised by $\omega_c$ and the $y$--coordinate by $R_c = v_0/\omega_c$. The initial Lorentz-factor of the particles is $\gamma_0=10$. The curve labeled "Exact" denotes the exact solution which is reproduced by the pushers Vay, HC, PL, LiLF, GH, GH2, and IMP. The curve for ZZ is not fully shown.
• Figure 2: The $y$--coordinate of the particle trajectory orbiting in a crossed electric and magnetic field (test case C) for a selection of schemes. The time steps are $\Delta t = 4\times10^{-4} T_g$ (left) and $2\times10^{-3} T_g$ (right)
• Figure 3: Poincaré plots for the particle orbits in parallel $\mathbf{E}$ and $\mathbf{B}$ fields with spatial variation, described in section \ref{['sec:hg_test']}. The time step is $\Delta t = T_{\text{osc}}/40$. The remaining parameters are given in the main text. Only the results for the Boris scheme (left) and the PL scheme (right) are shown.
• Figure 4: Scaling of the numerical error in the canonical momentum $p_z$ with the time step for the particle orbits in parallel $\mathbf{E}$ and $\mathbf{B}$ fields with spatial variation, described in section \ref{['sec:hg_test']}. GH and GH2 produce identical results. IMP is not shown because the error is zero within machine precision.
• Figure 5: Numerical errors $\mathcal{E}^{\mathrm{Wave}}_{L}$ (left) and $\mathcal{E}^{\mathrm{Wave}}_{E}$ (right) of the calculated motion in a relativistic plane wave (test case F) as a function of the time step $\Delta t$. $\mathcal{E}^{\mathrm{Wave}}_{L}$ is within machine precision for PL, LiLF, GH, GH2, and IMP. $\mathcal{E}^{\mathrm{Wave}}_{E}$ curves for CC and GYR are not shown because they are similar to Boris and Vay respectively.