A simple modification of the artificial bulk viscosity is proposed for LES of compressible turbulent flows with shocks. The proposed bulk viscosity shows significant improvements in the capability of the method for simulating turbulent flows involving shocks while retaining the shock-capturing capability. The Ducros-type switching used in the model removes the unnecessary bulk viscosity at weak compression regions and allows the bulk viscosity to localize only near the shocks. It also eliminates the need for any wall-damping function for the bulk viscosity while simulating wall-bounded turbulent flows. Detailed results from the evolution of decaying compressible isotropic turbulence with eddy-shocklets and supersonic turbulent boundary layers are used in the assessments made in this paper. Finally an illustrative application to the problem of mixing of a choked jet injected into a supersonic crossflow is given using the numerical scheme developed here. The compact differencing and filtering schemes coupled with localized artificial diffusivity methodology without adding an explicit SGS model is seen to perform better than with the inclusion of a SGS model. Inclusion of a SGS model in addition to the low-pass filtering and artificial bulk viscosity introduced additional dissipation. However, investigations at higher Reynolds numbers need to be considered to evaluate this issue further. Implicit LES approach with a pure 7th-order WENO scheme is found to be too dissipative. The flow statistics obtained using the second-order implicit time-integration scheme with multiple subiterations closely agrees with the explicit scheme. Thus, the implicit time-integration scheme is a good compromise in terms of accuracy and efficiency.