The second order and higher order (in time) accurate numerical schemes are studied for a class of bi-stable gradient flows, such as the standard Cahn-Hilliard equation and the epitaxial thin film growth models. The convex splitting nature indicates their unique solvability and unconditional energy stability. For the second order temporal approximation, both the modified Crank-Nicholson and the backward differentiation formula (BDF) versions are considered. Moreover, a linearized approach of the nonlinear part is proposed, and the energy stability could be assured with a careful analysis. As a result of the energy stability, a sharper error estimate with an improved convergence constant available for these proposed numerical scheme. More tricky ideas could be applied to derive third order numerical schemes with a local in time convergence property. Energy stable third order numerical schemes are also under consideration.

Cheng Wang received his Ph.D (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; he has had over 40 peer-reviewed articles at different scientific journals.