The Consistent Multi-fluid Advection method
  Simple modifications for higher-order Godunov-type difference schemes are presented which allow for accurate advection of multi-fluid flows in hydrodynamic simulations. The constraint that the sum of all mass fractions has to be equal to one in every computational zone throughout the simulation is fulfilled by renormalizing the mass fractions during the advection step. The proposed modification is appropriate for any difference scheme written in conservation form. Unlike other commonly used methods it does not violate the conservative character of the advection method. A new steepening mechanism, which is based on modification of interpolation profiles, is used to reduce numerical diffusion across composition discontinuities. Additional procedures are described, which are necessary to enforce monotonicity. Several numerical experiments are presented which demonstrate the capability of our Consistent Multi-fluid Advection (CMA) method in case of smooth and discontinuous distributions of fluid phases and under different hydrodynamic conditions. It is shown that due to the reduced diffusivity of the proposed scheme the abundance of some heavy elements obtained from hydrodynamic simulations of type II supernova explosions can change by a factor of a few in the most extreme cases. 
Tomasz Plewa and Ewald Mueller (MPA, Garching)