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A reduced-speed-of-light (RSOL) approximation is a useful technique formagnetohydrodynamic (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 theCourant condition given the large CR speeds). However, due to the absence of awell-formulated RSOL implementation from the literature, with naivesubstitution of the true $c$ with a RSOL, the CR properties in MHD-PICsimulations (e.g.\ CR energy or momentum density, gyro radius) varyartificially 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 functionand 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 ofvarious terms like the electric field, and (4) ensures the numerical timestepfor CRs can always be safely increased by a factor $\sim c/\tilde{c}$. This newRSOL formulation should enable greater self-consistency and reduced CPU cost insimulations of CR-MHD interactions.

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