A Reduced-Speed-of-Light Formulation of the Magnetohydrodynamic-Particle-in-Cell Method

A reduced-speed-of-light (RSOL) approximation is a useful technique for magnetohydrodynamic (MHD)-particle-in-cell (PIC) simulations. 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 from the literature, with naive substitution of the true $c$ with a RSOL, the CR properties in MHD-PIC 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 choices of a RSOL. 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 {\em independent} of the RSOL $\tilde{c}$ even for $\tilde{c} \ll c$, (2) ensures that the simulation can {\em 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 new RSOL formulation should enable greater self-consistency and reduced CPU cost in simulations of CR-MHD interactions.