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We present the first plasma simulations obtained with the code dHybridR, ahybrid particle-in-cell code with fluid electrons and both thermal andenergetic ions that retain relativistic dynamics. dHybridR is constructed tostudy astrophysical and space-physics problems where a few energeticnon-thermal particles (i.e., cosmic rays, CRs) affect the overall dynamics of anon-relativistic plasma, such as CR-driven instabilities, collisionless shocks,magnetic reconnection, turbulence, etc.In this method paper we provide someapplications to linear (resonant/non-resonant CR streaming instability) andstrongly non-linear (parallel shocks) problems that show the capabilities ofthe code. In particular, we provide the first self-consistent hybrid runs thatshow the acceleration of relativistic ions at non-relativistic shocks; CRsdevelop a power-law in momentum, which translates to a broken power law inenergy that exhibits a steepening around the ion rest mass, as predicted by thetheory of diffusive shock acceleration. We present examples of 2D dHybridR runsrelevant for fast shocks in radio supernovae, whose evolution can be followedin real time, and 3D runs of low-Mach-number heliospheric shocks, which can becompared with in-situ spacecraft observations.

dHybridR: a Hybrid--Particle-in-Cell Code Including Relativistic Ion Dynamics