In this talk we describe two fully mass conservative, energy stable, finite difference methods on a staggered grid for the quasi-incompressible Navier-Stokes-Cahn-Hilliard (q-NSCH) system governing a binary incompressible fluid flow with variable density and viscosity. Both methods, namely the primitive method (finite difference method in the primitive variable formulation) and the projection method (finite difference method in a projection-type formulation), are so designed that the mass of the binary fluid is preserved, and the energy of the system equations is always non-increasing in time at the fully discrete level. We also present an efficient, practical nonlinear multigrid method - comprised of a standard FAS method for the Cahn-Hilliard equation, and a method based on the Vanka-type smoothing strategy for the Navier-Stokes equation - for solving these equations. We test the scheme in the context of Capillary Waves, rising droplets and Rayleigh-Taylor instability. Quantitative comparisons are made with existing analytical solutions or previous numerical results that validate the accuracy of our numerical schemes. Moreover, in all cases, mass of the single component and the binary fluid was conserved up to 10^{－8} and energy decreases in time. This is a joint work with Zhenlin Guo, John Lowengrub and Steven Wise.

Ping Lin obtained BSc in Mathematics in Nanjing University (China) and a PhD  in Applied Mathematics from University of British Columbia in 1996.  After a couple of postdoctoral positions in Stanford University and Rensselaer Polytechnic Institute, he took up an assistant professor position at the National University of Singapore in 1999 and then promoted to associate professor and full professor. He took up the chair of numerical analysis position in University of Dundee in 2007.  The main focus of his research is computational and applied mathematics and mathematical modeling. His research has primarily been concerned with theoretical and numerical analysis, computation and various applications of ODEs and PDEs, possibly with constraints. Applications include Newtonian and non-Newtonian fluids, multi-phase flows, atomistic-to-continuum coupling in solids, some biological and medical problems, and image processing.