Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy

• Authors: Jie Shen;Cheng Wang;Xiaoming Wang;Steven M. Wise
• Venue: SIAM Journal on Numerical Analysis
• Year: 2012

We construct unconditionally stable, unconditionally uniquely solvable, and second-order accurate (in time) schemes for gradient flows with energy of the form $\int_\Omega ( F(\nabla\phi({\bf x})) + \frac{\epsilon^2}{2}|\Delta\phi({\bf x})|^2 ) d{\bf x}$. The construction of the schemes involves the appropriate combination and extension of two classical ideas: (i) appropriate convex-concave decomposition of the energy functional and (ii) the secant method. As an application, we derive 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.