There are two fundamental numerical challenges associated with solving fluid flow problems involving multiple fluids/components/phases/scales: (1) solving PDEs with discontinuous coefficients and interface conditions; (2) evolving in time the geometry, e.g., the interface between the various components. In this talk, the speaker will present high-order numerical techniques to solve these problems on a regular Cartesian grid. First, the speaker will introduce the Correction Function Method (CFM) framework, and will apply it to solving Poisson’s equation with interface jump discontinuities, a canonical problem. Second, the speaker will introduce the Gradient-Augmented Level Set Method (GALSM), and will apply it to the problem of evolving interfaces separating the various fluid domains. Throughout this talk, the speaker will illustrate his approach with simulations of physical systems. The speaker will end by showing a surprising extension of the methods developed to solve with arbitrary resolution the incompressible Euler equations.