Charm mixing and $CP$ violation at LHCb

LHCb collected the world's largest sample of open charm decays during Run 1 of the Large Hadron Collider. This has permitted many precision measurements of charm mixing and $CP$ violation parameters, the most precise of which being $\Delta A_{CP}$, a measurement of the relative strength of direct, time-integrated $CP$ asymmetries between two singly-Cabibbo suppressed $D^{0}$ decays. This measurement has recently been updated using promptly-produced $D^{0}$ mesons with the full Run 1 dataset, and has a precision below the per mille level. In addition, LHCb has recently made the first observation of $D^{0}$ mixing in a multi-body $D^{0}$ decay, also measuring associated coherence parameters which can be used as input to measurements of the CKM angle $\gamma$. LHCb has also measured the mixing parameters $x$ and $y$ with a model-independent analysis of $D^{0} \to K_{S}^{0}\pi^{+}\pi^{-}$ decays, and the size of direct $CP$ violation in $D^{0} \to K_{S}^{0}K_{S}^{0}$ decays. These four analyses will be presented, along with a brief overview of the prospects for Run 2.