Large-eddy simulation (LES) of an under-expanded sonic jet injection into a supersonic turbulent crossflow (JISC) is performed to obtain insights into the physics of the jet mixing. Recently developed localized artificial diffusivity scheme for shock capturing with high-order compact scheme is used. Progressive mesh refinement study is conducted to quantify the broadband range of scales of turbulence which are resolved in the present calculations. The calculations reported aim to reproduce the flow conditions reported in the experiments of Santiago and Dutton [J. Prop. Power. 13 (1997) 264-273] and detailed comparison with these data are shown. Jet mixing calculations where the upstream boundary layer is fully turbulent are also compared with corresponding calculations with a nominally laminar boundary layer. Statistics obtained by the LES with turbulent crossflow showed good agreement with the experiment and a series of mesh refinement study showed reasonable grid convergence in the predicted mean and fluctuation flow quantities. The present LES also reproduce the large-scale dynamics of the flow reported in previous experiments, but the richness of data provided by LES allows a much deeper exploration of the flow physics. Key physics of the jet mixing in a supersonic crossflow are highlighted by exploring the physics underlying the observed unsteady phenomena. Effect of approaching turbulent boundary layer on the jet mixing is investigated by comparing the results of JISC with turbulent and laminar crossflow.