The Richtmyer-Meshkov instability (RMI) and the subsequent turbulent mixing driven by the interaction of shock waves with interfaces separating materials of different densities are commonly found in many natural phenomena and engineering applications with high-speed flows. One of the goals in this thesis is to develop accurate and efficient numerical methods that are suitable for numerical simulations of this kind of flows that involve both shock waves and turbulent motions. A type of high-order shock-capturing schemes that can be in explicit or spatially implicit form is developed to achieve this goal with localized dissipation nonlinear weighting technique. The scheme has the ability to preserve fine-scale features in smooth regions with minimal dissipation while still has the ability to provide sufficient numerical dissipation to capture shocks and discontinuities robustly. The explicit form of the high-order scheme is implemented in an in-house adaptive mesh refinement (AMR) framework which can efficiently employ the computational resources by dynamically allocating fine grid cells only to regions containing features of interest for multi-species Navier-Stokes simulations. As another goal of this thesis, the AMR framework is used to conduct two-dimensional (2D) and three-dimensional (3D) high-resolution simulations for the study of the RMI-induced mixing emerging from the interaction between a Mach 1.45 shock wave and a perturbed planar interface between sulphur hexafluoride and air. The numerical results are used to examine the differences between the development of RMI in 2D and 3D configurations during two different stages: (1) initial growth of hydrodynamic instability from multi-mode perturbations after the arrival of primary shock and (2) transition to chaotic or turbulent state after re-shock. The effects of the Reynolds number on the mixing in 3D simulations are also studied through varying the transport coefficients. An analysis of second-moment budgets for the highest Reynolds number 3D case is also performed. The analysis first addresses the importance of the second moment quantities: turbulent mass flux and density-specific-volume covariance for the closure of Favre-averaged Navier--Stokes (FANS) equations in this type of flow compared to single-species incompressible flows that only require Reynolds stresses for closure. The budgets of different second-moments before and after re-shock are also studied and compared in details. Further analysis is conducted on the post-transition flow to examine the validity of the modeling assumptions in the Besnard-Harlow-Rauenzahn-3 model and its variants for the unclosed terms in the FANS equations.