Multi-component flows occur in many areas of Physics and Engineering. The numerical simulation of such problems usually involves tracking one or more level set functions, used to define the interfaces between the components implicitly. A challenge in this formulation is how to correctly state boundary conditions across the different regions (possibly involving different PDEs). Another challenge is how to represent and track small (sub-grid) structures without loss of mass. After some motivating examples (falling liquid films, solid/fluid interaction, bubbles on soap films, and thin filaments in complex fluids) we will present a new method for solving the advection equation for a level set function. The approach relies on carrying both function values and gradients of the level set function as coupled evolved quantities, using Hermite interpolants and a CIR (Courant–Isaacson–Rees) strategy. This allows for improved mass conservation, removes the need to solve a reinitialization equation, and allows capturing features smaller than the grid resolution on a regular Eulerian mesh. For the specific choice of bi (tri)-cubic Hermite interpolation we obtain a stable globally 3rd order method. We will discuss some advantages of the proposed approach: locality, easy computation of curvature and surface normals, application to quadtree/octree structures, and possible extensions to higher order.