The ability of a finite volume Godunov and a semi-Lagrangian large-eddy simulation (LES) method to predict shock induced turbulent mixing has been examined through simulations of the half-height experiment [Holder and Barton. In: Proceedings of the international workshop on the physics of compressible turbulent mixing, 2004]. Very good agreement is gained in qualitative comparisons with experimental results for combined Richtmyer-Meshkov and Kelvin-Helmholtz instabilities in compressible turbulent multi-component flows. It is shown that both numerical methods can capture the size, location and temporal growth of the main flow features. In comparing the methods, there is variability in the amount of resolved turbulent kinetic energy. The semi-Lagrangian method has constant dissipation at low Mach number, thus allowing the initially small perturbations to develop into Kelvin-Helmholtz instabilities. These are suppressed at the low Mach stage in the Godunov method. However, there is an excellent agreement in the final amount of fluid mixing when comparing both numerical methods at different grid resolutions.