Developing a highly accurate numerical framework to study multiphase mixing in high speed flows containing shear layers, shocks, and strong accelerations is critical to many scientific and engineering endeavors. These flows occur across a wide range of scales: from tiny bubbles in human tissue to massive stars collapsing. The lack of understanding of these flows has impeded the success of many engineering applications, our comprehension of astrophysical and planetary formation processes, and the development of biomedical technologies. We present advances in the three fields of numerical methods, high performance computing, and multiphase flow modeling: (i) novel Discontinuous Galerkin numerical methods to capture accurately the multiphase nature of the problem; (ii) modern high performance computing paradigms using the Message Passing Interface and Graphics Processing Units to resolve the disparate time and length scales of the physical processes; (iii) new insights and models of the dynamics of multiphase flows, including mixing through hydrodynamic instabilities.