Generalized, energy-conserving numerical simulations of particles in general relativity. II. Test particles in electromagnetic fields and GRMHD

Direct observations of compact objects, in the form of radiation spectra, gravitational waves from VIRGO/LIGO, and forthcoming direct imaging, are currently one of the primary source of information on the physics of plasmas in extreme astrophysical environments. The modeling of such physical phenomena requires numerical methods that allow for the simulation of microscopic plasma dynamics in presence of both strong gravity and electromagnetic fields. In Bacchini et al. (2018) we presented a detailed study on numerical techniques for the integration of free geodesic motion. Here we extend the study by introducing electromagnetic forces in the simulation of charged particles in curved spacetimes. We extend the Hamiltonian energy-conserving method presented in Bacchini et al. (2018) to include the Lorentz force and we test its performance compared to that of standard explicit Runge-Kutta and implicit midpoint rule schemes against analytic solutions. Then, we show the application of the numerical schemes to the integration of test particle trajectories in general relativistic magnetohydrodynamic (GRMHD) simulations, by modifying the algorithms to handle grid-based electromagnetic fields. We test this approach by simulating ensembles of charged particles in a static GRMHD configuration obtained with the Black Hole Accretion Code (BHAC)

Magnetohydrodynamic-Particle-in-Cell Method for Coupling Cosmic Rays with a Thermal Plasma: Application to Non-relativistic Shocks

We formulate a magnetohydrodynamic-particle-in-cell (MHD-PIC) method for describing the interaction between collisionless cosmic ray (CR) particles and a thermal plasma. The thermal plasma is treated as a fluid, obeying equations of ideal MHD, while CRs are treated as relativistic Lagrangian particles subject to the Lorentz force. Backreaction from CRs to the gas is included in the form of momentum and energy feedback. In addition, we include the electromagnetic feedback due to CR-induced Hall effect that becomes important when the electron-ion drift velocity of the background plasma induced by CRs approaches the Alfv\'en velocity. Our method is applicable on scales much larger than the ion inertial length, bypassing the microscopic scales that must be resolved in conventional PIC methods, while retaining the full kinetic nature of the CRs. We have implemented and tested this method in the Athena MHD code, where the overall scheme is second-order accurate and fully conservative. As a first application, we describe a numerical experiment to study particle acceleration in non-relativistic shocks. Using a simplified prescription for ion injection, we reproduce the shock structure and the CR energy spectra obtained with more self-consistent hybrid-PIC simulations, but at substantially reduced computational cost. We also show that the CR-induced Hall effect reduces the growth rate of the Bell instability and affects the gas dynamics in the vicinity of the shock front. As a step forward, we are able to capture the transition of particle acceleration from non relativistic to relativistic regimes, with momentum spectrum $f(p)\sim p^{-4}$ connecting smoothly through the transition, as expected from the theory of Fermi acceleration.Comment: 24 pages, 15 figures, accepted for publication in Ap

Leapfrog methods for relativistic charged-particle dynamics

A basic leapfrog integrator and its energy-preserving and variational / symplectic variants are proposed and studied for the numerical integration of the equations of motion of relativistic charged particles in an electromagnetic field. The methods are based on a four-dimensional formulation of the equations of motion. Structure-preserving properties of the numerical methods are analysed, in particular conservation and long-time near-conservation of energy and mass shell as well as preservation of volume in phase space. In the non-relativistic limit, the considered methods reduce to the Boris algorithm for non-relativistic charged-particle dynamics and its energy-preserving and variational / symplectic variants.Comment: 18 pages, 3 figure