Accurately Incorporating a Reduced-Speed-of-Light in
  Magnetohydrodynamic-Particle-in-Cell Simulations
  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 $\tilde{c}$ 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 ($\tilde{c} \rightarrow c$) 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 $\tilde{c}$ even for $\tilde{c} \ll c$, (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 $\sim c/\tilde{c}$.
This should enable greater self-consistency and reduced CPU cost in simulations
of CR-MHD interactions.
