Abstract: Some of the most interesting problems in nature are those involving multiple interacting scales. To simulate these systems, one must often make the painful choice of resolving some range of scales while ignoring or approximating the rest. Classical examples of this are the simulation of turbulence, high Schmidt/Prandtl number transfer processes, or phase-field models.   In the simulation of multi-component fluids, this multi-scale numerical challenges are (1) solving PDEs with discontinuous coefficients and interface conditions, (2) evolving in time a geometric object e.g. density, concentration, the interface between two phases …   In this talk I will present numerical techniques to address these problems on a regular Cartesian grid. I will end with a surprising extension of these methods developed to solve the 2D incompressible Euler equations with arbitrary resolution.