Unconditionally stable schemes for equations of thin film epitaxy

Chen Wang, Xiaoming Wang, Steven M. Wise

We present unconditionally stable and convergent numerical schemes for gradient flows with energy of the form $\int_\Omega \left( F(\nabla\phi(\bx)) + \frac{\epsilon^2}{2}|\Delta\phi(\bx)|^2 \right)\,\mathrm{d}\bx$. The construction of the schemes involves an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. As an application, we derive unconditionally stable and convergent schemes for epitaxial film growth models with slope selection ($F\left(\by\right)= \frac14(\left|\by\right|^2-1)^2$) and without slope selection ($F\left(\by\right)=-\frac{1}{2}\ln(1+\left|\by\right|^2)$). We conclude the paper with some preliminary computations that employ the proposed schemes.