Stability Analysis of Large Time‐Stepping Methods for Epitaxial Growth Models

Abstract

Numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy (MBE) require very large time simulation, and therefore large time steps become necessary. The main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit-explicit approach. To this end, an extra term, which is consistent with the order of the time discretization, is added to stabilize the numerical schemes. Then the stability properties of the resulting schemes are established rigorously. Numerical experiments are carried out to support the theoretical claims. The proposed methods are also applied to simulate the MBE models with large solution times. The power laws for the coarsening process are obtained and are compared with previously published results. Introduction.Recently there has been significant research interest in the dynamics of molecular beam epitaxy (MBE) growth. The MBE technique is among the most refined methods for the growth of thin solid films, and it is of great importance for applied studies; see, e.g., [1,16,22]. The evolution of the surface morphology during epitaxial growth results in a delicate relation between the molecular flux and the relaxation of the surface profile through surface diffusion of adatoms. It occurs on time and length scales that may span several orders of magnitude. Different kinds of models have been used to describe such phenomena; these typically include atomistic models, continuum models, and hybrid models. The atomistic models are usually implemented in the form of molecular dynamics or kinetic Monte Carlo simulations [4,9,17]. The continuum models are based on partial differential equations and are appropriate mainly for investigating the temporal evolution of the MBE instability at large time and length scales [11,24]. The hybrid models can be considered as a compromise between atomistic models and continuum models; see, e.g., [3,8].We are interested in the continuum models for the evolution of the MBE growth. Let h(x, t) be the epitaxy surface height with x ∈ R 2 and t ≥ 0. Under typical conditions for MBE growth, the height evolution equation can be written under mass conservation form (see, e.g., [14]):

Simulating Surface-Mediated Self Assembly Patterns by a Stabilized Fourier Spectral Method

“…Our numerical results show later that the method processes better stability properties. Similar stabilization procedures were also used by Zhu et al 22) for variablemobility coarsening kinetics simulation and the approach was presented by Xu and Tang 21) for large time-stepping simulation of molecular beam epitaxy growth. Rigorous mathematical proof and stability analysis of this class of stabilization schemes can be found in.…”

“…Rigorous mathematical proof and stability analysis of this class of stabilization schemes can be found in. 21) 4. Results and Discussion 4.1 Self-assembled patterns in epilayers with different average concentrations In order to validate the correctness and particularly the numerical stability of the proposed scheme for self-assembly simulation, self-assembly in epilayers with different average concentrations on a solid substrate are simulated based on the proposed method.…”

“…Following Xu and Tang, 21) we add a stabilization term and adopt a second-order time discretization scheme, and eventually present an improved algorithm as follows…”

Simulação tridimensional adaptativa da separação das fases de uma mistura bifásica usando a equação de Cahn-Hilliard

“…Adotamos mobilidade constante, isto é, M (φ e (t, x)) = 1, condições de contorno periódicas e o passo temporal (3.8). Para as constantes presentes nas equações discretas (3.6) e (3.7), atribuimos os valoresM = 1, τ = 2 [22] e ǫ 2 = 0, 01. A Tabela 1 mostra os resultados obtidos para o teste em uma malha uniforme e na malha com refinamento local descrita anteriormente, respectivamente.…”

Modeling and Numerical Approximation of Two-Phase Incompressible Flows by a Phase-Field Approach

“…• To alleviate the difficulty associated with the stiffness caused by the thin interfacial width, a stabilizing term in the phase equation was introduced in [77,46] (see also [75]). This stabilizing term allows us to treat the nonlinear term in the phase equation explicitly without suffering from any time step constraint.…”