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We formulate a magnetohydrodynamic-particle-in-cell (MHD-PIC) method fordescribing the interaction between collisionless cosmic ray (CR) particles anda thermal plasma. The thermal plasma is treated as a fluid, obeying equationsof ideal MHD, while CRs are treated as relativistic Lagrangian particlessubject to the Lorentz force. Backreaction from CRs to the gas is included inthe form of momentum and energy feedback. In addition, we include theelectromagnetic feedback due to CR-induced Hall effect that becomes importantwhen the electron-ion drift velocity of the background plasma induced by CRsapproaches the Alfv\'en velocity. Our method is applicable on scales muchlarger than the ion inertial length, bypassing the microscopic scales that mustbe resolved in conventional PIC methods, while retaining the full kineticnature of the CRs. We have implemented and tested this method in the Athena MHDcode, where the overall scheme is second-order accurate and fully conservative.As a first application, we describe a numerical experiment to study particleacceleration in non-relativistic shocks. Using a simplified prescription forion injection, we reproduce the shock structure and the CR energy spectraobtained with more self-consistent hybrid-PIC simulations, but at substantiallyreduced computational cost. We also show that the CR-induced Hall effectreduces the growth rate of the Bell instability and affects the gas dynamics inthe vicinity of the shock front. As a step forward, we are able to capture thetransition of particle acceleration from non relativistic to relativisticregimes, with momentum spectrum $f(p)\sim p^{-4}$ connecting smoothly throughthe transition, as expected from the theory of Fermi acceleration.

Magnetohydrodynamic-Particle-in-Cell Method for Coupling Cosmic Rays with a Thermal Plasma: Application to Non-relativistic Shocks