An energy stable numerical scheme for the Cahn-Hilliard  equation is proposed and analyzed, with second order accuracy  in time and the fourth order finite difference approximation  in space. In particular, the truncation error for the long  stencil fourth order finite difference approximation is estimated,  over a uniform numerical grid with a periodic boundary condition,  via the help of discrete Fourier analysis instead of the the  standard Taylor expansion. This in turn results in a reduced  regularity requirement for the test function. In the temporal  approximation, we apply a second order BDF stencil, combined  with a second order extrapolation formula applied to the  concave diffusion term, as well as a second order artificial  Douglas-Dupont regularization term, for the sake of energy  stability. As a result, the unique solvability, energy stability  are established for the proposed numerical scheme, and an optimal  rate convergence analysis is derived. A few numerical experiments  are also presented in this talk.

Cheng Wang received his PhD (under Jian-Guo Liu) at Temple University, USA, in 2000. Afterward, he spent 3 years at Indiana University as a postdoc, with Roger Temam and Shouhong Wang as his postdoc mentors. In 2003-2008, he was a tenure-track assistant professor at University of Tennessee. In 2008, he moved to University of Massachusetts Dartmouth; right now, he is an associate professor there. His research area is applied mathematics, including numerical analysis, partial differential equations, fluid mechanics, computational electro-magnetics, material sciences, etc.