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

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 L2 gradient flow of the energy ∫Ω(-1/2 ln(1 + |ø|2) + ε2 2 |Ø(x)|2) dx. 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 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. © Springer Science+Business Media, LLC 2011

Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth

In the present contribution we review basic mathematical results for three physical systems involving self-organising solid or liquid films at solid surfaces. The films may undergo a structuring process by dewetting, evaporation/condensation or epitaxial growth, respectively. We highlight similarities and differences of the three systems based on the observation that in certain limits all of them may be described using models of similar form, i.e., time evolution equations for the film thickness profile. Those equations represent gradient dynamics characterized by mobility functions and an underlying energy functional. Two basic steps of mathematical analysis are used to compare the different system. First, we discuss the linear stability of homogeneous steady states, i.e., flat films; and second the systematics of non-trivial steady states, i.e., drop/hole states for dewetting films and quantum dot states in epitaxial growth, respectively. Our aim is to illustrate that the underlying solution structure might be very complex as in the case of epitaxial growth but can be better understood when comparing to the much simpler results for the dewetting liquid film. We furthermore show that the numerical continuation techniques employed can shed some light on this structure in a more convenient way than time-stepping methods. Finally we discuss that the usage of the employed general formulation does not only relate seemingly not related physical systems mathematically, but does as well allow to discuss model extensions in a more unified way

An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematic