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 …