Multiphase flows are ubiquitous in natural and engineered systems and occur over scales ranging many orders of magnitude, from the movement of the ocean to the flow of blood through a capillary. In each case, the interface between the phases plays a key role in the dynamics of the flow. In this dissertation, I will detail the development of novel numerical methods for accurately and efficiently tracking the interface in simulations of multiphase flows, specifically focusing on flow configurations involving large density ratios, high shear at the interface, and liquid structures that span orders of magnitude in size. One specific instance where this occurs is during primary atomization, where a large liquid structure breaks up into smaller, more stable structures. This important class of flows is present in many different industries that currently employ over 2.5 million people. Enabling more accurate simulation of primary atomization is a central motivation of my work.