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We consider a phase field model for molecular beam epitaxial growth with slope selection with the goal of determining linear energy stable time integration methods for the dynamics. Stable methods for this model have been found via a concave-convex splitting of the dynamics, but this approach generally leads to a nonlinear update equation. We seek a linear energy stable method to allow for simple and efficient time marching with fast Fourier transforms. Our approach is to parametrize a class of semi-implicit methods and perform unconditional von Neumann stability analysis to identify the region of stability in parameter space. Since unconditional von Neumann stability does not ensure energy stability, we perform extensive numerical tests and find strong agreement between the predicted and observed stable regions of parameter space. This analysis elucidates a novel feature that the stability region in parameter space differs for a mono-domain system (single equilibrium slope) versus a many-domain system (coarsening facets from an initially flat surface). The utility of these steps is then demonstrated by a comparison of the coarsening dynamics for isotropic and anisotropic variants of the model.

The temperature dependence of the dynamics of the fraction F(t) of persistent spins in the triangular Q-state Potts model is investigated by large scale Monte Carlo simulations. After extending Derrida's approach of measuring the fraction of spins that remain in one phase to allQ low-temperature phases, it is shown that the exponent θ of algebraic decay of F(t) is independent of temperature.

Mesh adaptation is a reliable and effective method to improve the precision of flow simulation with computational fluid dynamics. Mesh refinement is a common technique to simulate steady flows. In order to dynamically optimize the mesh for transient flows, mesh coarsening is also required to be involved in an iterative procedure. In this paper, we propose a robust mesh adaptation method, both refinement and coarsening included. A data structure of $k$-way tree is adopted to save and access the parent–children relationship of mesh elements. Local element subdivision is employed to refine mesh, and element mergence is devised to coarsen mesh. The unrefined elements adjacent to a refined element are converted to polyhedrons to eliminate suspending points, which can also prevent refinement diffusing from one refined element to its neighbors. Based on an adaptation detector for vortices recognizing, the mesh adaptation was integrated to simulate the unsteady flow around a tri-wedges. The numerical results show that the mesh zones where vortices located are refined in real time and the vortices are resolved better with mesh adaptation.

A consumer behavior model is considered in the context of a network of interacting individuals in an energy market. We propose and analyze a simple dynamical model of an ensemble of coupled active elements mimicking consumers' behavior, where "word-of-mouth" interactions between individuals are important. A single element is modeled using the automatic control system framework. Assuming local (nearest neighbor) coupling we study the evolution of chains and lattices of the model consumers when varying the coupling strength and initial conditions. The results are interpreted as the dynamics of the decision-making process by the energy-market consumers. We demonstrate that a pitchfork bifurcation to the homogeneous solution leads to bistability of stationary regimes, while the autonomous system is always monostable. In the presence of inhomogeneities, this results in the formation of clusters of sharply positive and negative opinions. We also find that, depending on the coupling strength, the perturbations caused by inhomogeneities can be exponentially localized in space or delocalized. In the latter case, the coarse-graining of opinion clusters occurs.

We study the homogenization limit of the Mullins–Sekerka problem which serves as a model for late-stage coarsening in a phase transformation. We consider the case that the screening length which describes the effective range of particle interactions is much smaller than the system size, which leads to homogenization problems in unbounded domains.

The present paper deals with deterministic initial particle distributions which are in an average sense homogeneously distributed. Stochastic particle distributions will be considered in a second paper (Part II).

X-rays are ideal to study the structure of crystals due to their weak interaction with matter and in most cases allow a quantitative analysis using kinematical theory. To study the incorporation of atoms during crystal growth and to analyze the kinetics on the crystal surface high primary beam intensities available at synchrotrons are required. Our studies of the molecular beam epitaxy growth of III–V semiconductors reveal that, despite their similarity in crystal structure, the surface kinetics of GaAs(001), InAs(001) and GaSb(001) differ strongly. GaAs shows an unexpectedly large coarsening exponent outside the predicted range of Ostwald ripening models during recovery. GaSb exhibits dramatically different surface morphology variations during growth and recovery. Overgrowth of GaAs by epitaxial MnAs demonstrates the ability of X-ray diffraction to follow an interface as it is buried during heteroepitaxy, which is not possible by reflection high-energy electron diffraction.

The coarsening of an array of vortex ripples prepared in an unstable state is discussed within the framework of a simple mass transfer model first introduced by K. H. Andersen et al. (see Ref. 1). Two scenarios for the selection of the final pattern are identified. When the initial state is homogeneous with uniform random perturbations, a unique final state is reached which depends only on the shape of the interaction function f(λ). A potential formulation of the dynamics suggests that the final wavelength is determined by a Maxwell construction applied to f(λ), but comparison with numerical simulations shows that this yields only an upper bound. In contrast, the evolution from a perfectly homogeneous state with a localized perturbation proceeds through the propagation of wavelength doubling fronts. The front speed can be predicted by standard marginal stability theory. In this case the final wavelength depends on the initial wavelength in a complicated manner which involves multiplication by powers of two and rational ratios such as 4/3.

The coarsening of an array of vortex ripples prepared in an unstable state is discussed within the framework of a simple mass transfer model first introduced by K. H. Andersen et al. (see Ref. 1). Two scenarios for the selection of the final pattern are identified. When the initial state is homogeneous with uniform random perturbations, a unique final state is reached which depends only on the shape of the interaction function f(λ). A potential formulation of the dynamics suggests that the final wavelength is determined by a Maxwell construction applied to f(λ), but comparison with numerical simulations shows that this yields only an upper bound. In contrast, the evolution from a perfectly homogeneous state with a localized perturbation proceeds through the propagation of wavelength doubling fronts. The front speed can be predicted by standard marginal stability theory. In this case the final wavelength depends on the initial wavelength in a complicated manner which involves multiplication by powers of two and rational ratios such as 4/3.