Second-Order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to thin film epitaxy

Jie Shen, Cheng Wang, Xiaoming Wang, Steven Wise

We construct unconditionally stable, uniquely solvable and second-order in time schemes for gradient flows with energy of the form $\int_\Omega \left( F(\nabla\phi(\bf{x})) + \frac{\epsilon^2}{2}|\Delta\phi(\bf{x})|^2 \right)\,d\bf{x}$. The construction of the schemes involves appropriate combination and extensions of two classical ideas: (i) appropriate convex-concave decomposition of the energy functional, and (ii) the secant method. As an application, we derive unconditionally stable, second-order in time schemes for epitaxial growth models with slope selection ($F(\bf{y})= \frac14(|\bf{y}|^2-1)^2$) or without slope selection ($F(\bf{y})=-\frac12\ln(1+|\bf{y}|^2)$). Two types of unconditionally stable uniquely solvable second order schemes are presented. The first type inherits the variational structure of the original continuous in time gradient flow while the second type does not preserve the variational structure. We present numerical simulations for the case with slope selection which verify well-known physical scaling laws for the long time coarsening process.