Accurately Incorporating a Reduced-Speed-of-Light in Magnetohydrodynamic-Particle-in-Cell Simulations

Abstract

A reduced-speed-of-light (RSOL) approximation is usually adopted in magnetohydrodynamic (MHD)-particle-in-cell (PIC) simulations, in which relativistic cosmic ray (CR) particles moving at nearly the speed of light c and background (non-relativistic) plasma are evolved concurrently. With a RSOL, some "in-code" speed-of-light c̃ is set to much lower values than the true c, allowing simulations to take larger timesteps (which are restricted by the Courant condition given the large CR speeds). However, due to the absence of a well-formulated RSOL implementation in the literature for MHD-PIC simulations, the CR properties in simulations (e.g.\ CR energy or momentum density, gyro radius) vary artificially with respect to each other and with respect to the converged (c̃ →c) solutions with different implementations of a RSOL, breaking the correspondence between simulation results and physical reality. Here, we derive a new formulation of the MHD-PIC equations with a RSOL, and show that (1) it guarantees all steady-state properties of the CR distribution function and background plasma/MHD quantities are independent of the RSOL c̃ even for c̃ ≪c, (2) ensures that the simulation can simultaneously represent the real physical values of CR number, mass, momentum, and energy density, (3) retains the correct physical meaning of various terms like the electric field, and (4) ensures the numerical timestep for CRs can always be safely increased by a factor ∼c/c̃ . This should enable greater self-consistency and reduced CPU cost in simulations of CR-MHD interactions.