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The gyro-resonant cosmic-ray (CR) streaming instability is believed to play acrucial role in CR transport, leading to growth of Alfv\'en waves at smallscales that scatter CRs, and impacts the interaction of CRs with the ISM onlarge scales. However, extreme scale separation ($\lambda \ll \rm pc$), lowcosmic ray number density ($n_{\rm CR}/n_{\rm ISM} \sim 10^{-9}$), and weak CRanisotropy ($\sim v_A/c$) pose strong challenges for proper numerical studiesof this instability on a microphysical level. Employing the recently developedmagnetohydrodynamic-particle-in-cell (MHD-PIC) method, which has uniqueadvantages to alleviate these issues, we conduct one-dimensional simulationsthat quantitatively demonstrate the growth and saturation of the instability inthe parameter regime consistent with realistic CR streaming in the large-scaleISM. Our implementation of the $\delta f$ method dramatically reduces Poissonnoise and enables us to accurately capture wave growth over a broad spectrum,equally shared between left and right handed Alfv\'en modes. We are also ableto accurately follow the quasi-linear diffusion of CRs subsequent to wavegrowth, which is achieved by employing phase randomization across periodicboundaries. Full isotropization of the CRs in the wave frame requires pitchangles of most CRs to efficiently cross $90^\circ$, and can be captured insimulations with relatively high wave amplitude and/or high spatial resolution.We attribute this crossing to non-linear wave-particle interaction (rather thanmirror reflection) by investigating individual CR trajectories. We anticipateour methodology will open up opportunities for future investigations thatincorporate additional physics.

Magnetohydrodynamic-Particle-in-Cell Simulations of the Cosmic-Ray Streaming Instability: Linear Growth and Quasi-linear Evolution