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 second-order accurate (in time) schemes for gradient flows with energy of the form integral Omega(F(del phi(x))+epsilon(2)/2 vertical bar Delta phi(x)vertical bar(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 (vertical bar y vertical bar(2) - 1)(2)) or without slope selection (F(y) = -1/2 ln(1 + vertical bar y vertical bar(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.