SECOND-ORDER CONVEX SPLITTING SCHEMES FOR GRADIENT FLOWS WITH EHRLICH-SCHWOEBEL TYPE ENERGY: APPLICATION TO THIN FILM EPITAXY

• Published in: SIAM journal on numerical analysis 2012-01, Vol.50 (1), p.105-125
• Main Authors: SHEN, JIE, WANG, CHENG, WANG, XIAOMING, WISE, STEVEN M.
• Format: Article
• Language: English
• Subjects: Applied mathematics; Approximation; Convexity; Energy; Energy policy; Epitaxy; Finite difference methods; Growth models; Secant function; Secant lines; Thin films
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/110822839

We construct unconditionally stable, unconditionally uniquely solvable, and secondorder accurate (in time) schemes for gradient flows with energy of the form ∫ Ω (F(∇ø(×)) + $\frac{{{ \in ^2}}} {2}$ |∆ø(×)|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) = $\frac{1} {4}$ (|y|² - 1)²) or without slope selection (F(y) = - $\frac{1} {2}$ 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.