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 an RSOL, some 'in-code' speed-of-light (c) over tilde is set to much lower values than the true c, allowing simulations to take larger time-steps (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 ((c) over tilde -> c) solutions, with different choices of a RSOL. Here, we derive a new formulation of the MHD-PIC equations with an 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 (c) over tilde even for (c) over tilde << c; (2) it ensures that the simulation can simultaneously represent the real physical values of CR number, mass, momentum, and energy density; (3) it retains the correct physical meaning of various terms like the electric field; and (4) it ensures the numerical time-step for CRs can always be safely increased by a factor similar to c/(c) over tilde. This new RSOL formulation should enable greater self-consistency and reduced CPU cost in simulations of CR-MHD interactions.