A Linear Energy Stable Scheme for a Thin Film Model Without Slope Selection

Abstract

We present a linear numerical scheme for a model of epitaxial thin film growth without slope selection. The PDE, which is a nonlinear, fourth-order parabolic equation, is the L ² gradient flow of the energy $\int_{\Omega}( - \frac{1}{2} \ln(1 + |\nabla\phi|^{2} ) + \frac{\epsilon^{2}}{2}|\Delta\phi(\mathbf{x})|^{2})\,\mathrm{d}\mathbf{x}$. The idea of convex-concave decomposition of the energy functional is applied, which results in a numerical scheme that is unconditionally energy stable, i.e., energy dissipative. The particular decomposition used here places the nonlinear term in the concave part of the energy, in contrast to a previous convexity splitting scheme. As a result, the numerical scheme is fully linear at each time step and unconditionally solvable. Collocation Fourier spectral differentiation is used in the spatial discretization, and the unconditional energy stability is established in the fully discrete setting using a detailed energy estimate. We present numerical simulation results for a sequence of ϵ values ranging from 0.02 to 0.1. In particular, the long time simulations show the −log(t) decay law for the energy and the t ^1/2 growth law for the surface roughness, in agreement with theoretical analysis and experimental/numerical observations in earlier works.