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

We construct unconditionally stable, unconditionally uniquely solvable, and secondorder accurate (in time) schemes for gradient flows with energy of the form ∫ Ω(F (∇φ(x))+ 2 2 |Δφ(x)|2) dx. 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 (y) = 1 4 (|y|2 − 1)2) or without slope selection (F (y) = − 1 2 ln(1 + |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.