In this chapter, we present a high-order numerical methodology based on Discontinuous Galerkin schemes for the accurate simulation of shock-turbulence interactions. The method combines a positivity-preserving limiter that guarantees the robustness of simulations, an artificial diffusion operator that suppresses shock oscillations, and a vortex sensor that preserves the artificial diffusion to impact turbulent eddies. The ability of the method to represent accurately turbulence with a robust shock capture is demonstrated from simulations of the compressible Taylor-Green vortex at Mach 1.25 and Reynolds 1600.