Damped Kuramoto-Sivashinsky Equation Research Articles

Solving 2D damped Kuramoto-Sivashinsky with multiple relaxation time lattice Boltzmann method

Lattice Boltzmann method (LBM) is a powerful fluid flow solver. Using this method to solve other PDEs might be a difficult task. The first challenge is to find a suitable local equilibrium distribution function (EDF) capable of recovering the desired PDE. The next difficulty arises from the explicit nature of LBM. The conditional stability of the LBM algorithm affects the numerical solution accuracy. Damped Kuramoto–Sivashinsky (DKS) equation is a fourth–order PDE that recently challenged many numerical methods' abilities. This equation is highly sensitive to a parameter that causes three states of solutions in a small interval of freedom. In this paper, we challenged LBM to solve the two–dimensional DKS equation by finding EDF using the Chapman–Enskog analysis up to the fourth–order. Also, the von Neumann analysis and a simple genetic algorithm are applied to find reliable values for the free parameters. Furthermore, a modification on image-based ghost node method is proposed for implementation of the boundary conditions in the complex geometries.

Two-dimensional simulation of the damped Kuramoto–Sivashinsky equation via radial basis function-generated finite difference scheme combined with an exponential time discretization

We apply a numerical scheme based on a meshless method in space and an explicit exponential Runge-Kutta in time for the solution of the damped Kuramoto–Sivashinsky equation in two-dimensional spaces. The proposed meshless method is radial basis function-generated finite difference, which approximates the derivatives of the unknown function with respect to the spatial variables by a linear combination of the function values at given points in the domain and weights. Also, in this approach there is no need a mesh or triangulation for approximation. For each point, the weights are computed separately in its local sub-domain by solving a small radial basis function interpolant. Besides, a numerical algorithm based on singular value decomposition of the local radial basis function interpolation matrix [59] is applied to find the suitable shape parameter for each interpolation problem. We also consider an explicit time discretization based on exponential Runge–Kutta scheme such that its stability region is bigger than the classical form of Runge-Kutta method. Some numerical simulations are provided on the square, circular and annular domains to show the capability of the numerical scheme proposed here.

Transition from Markovian to Glassy Dynamics in Damped Kuramoto-Sivashinsky Turbulence

In this research, we study Damped Kuramoto-Sivashinsky (DKS) equation, one of the famous nonlinear differential equation which exhibits turbulence. To solve DKS equation, we use Exponential Time Differencing (ETD) scheme which is combined with the Pseudospectral method. To know the effect of damping factor term, we analyze the resulting dynamics using Lyapunov exponent and total autocorrelation function. From that analysis, we elucidate the transition of Markovian to glassy dynamics in DKS equation. We conclude that there is a kind of mixing of several modes in the dynamics of the DKS equation which resulting the glassy dynamics in this system.

Nano-patterning of surfaces by ion sputtering: Numerical study of the anisotropic damped Kuramoto-Sivashinsky equation

Nonlinear models for pattern evolution by ion beam sputtering on a material surface present an ongoing opportunity for new numerical simulations. A numerical analysis of the evolution of preexisting patterns is proposed to investigate surface dynamics, based on a 2D anisotropic damped Kuramoto-Sivashinsky equation, with periodic boundary conditions. A finite-difference semi-implicit time splitting scheme is employed on the discretization of the governing equation. Simulations were conducted with realistic coefficients related to physical parameters (anisotropies, beam orientation, diffusion). The stability of the numerical scheme is analyzed with time step and grid spacing tests for the pattern evolution, and the Method of Manufactured Solutions has been used to verify the proposed scheme. Ripples and hexagonal patterns were obtained from a monomodal initial condition for certain values of the damping coefficient, while spatiotemporal chaos appeared for lower values. The anisotropy effects on pattern formation were studied, varying the angle of incidence of the ion beam with respect to the irradiated surface. Analytical discussions are based on linear and weakly nonlinear analysis.

Role of atom redeposition during rising ion flux in ion-induced nanodot self-assembly on GaSb surfaces

In this study, a bottom-up approach of ion irradiation from hot cathode DC discharge plasma was used to investigate the role of energetic ion flux on the self-assembly of GaSb nanodots (NDs) at normal incidence. It was observed that, when increasing the flux in the range of 1014–1015 ions cm−2 s−1, the lateral dimension and root mean square (RMS) roughness of NDs is reduced even at constant temperature conditions in the ion energy range from 400–800 eV. The evolution of the surface morphology for different flux regimes is observed in a numerical integration simulation using the nonlinear isotropic damped Kuramoto–Sivashinsky (DKS) equation. By introducing a redeposition term, the DKS equation is found to be in good qualitative agreement with the experimental results. We have demonstrated the linear dependency of the redeposition coefficient on the ion flux and also reported the nonlinear dependency on the thermal diffusion coefficient, transition time, and characteristic length with the flux. In accordance with the nonlinearity, we have also discussed the effect of the variation of the ion flux on the RMS roughness and lateral dimension of NDs.

Stability and symmetry of ion-induced surface patterning

We present a continuum model of ion-induced surface patterning. The model incorporates the atomic processes of sputtering, re-deposition and surface diffusion, and is shown to display the generic features of the damped Kuramoto-Sivashinsky (KS) equation of non-linear dynamics. Linear and non-linear stability analyses of the evolution equation give estimates of the emerging pattern wavelength and spatial symmetry. The analytical theory is confirmed by numerical simulations of the evolution equation with the Fast Fourier Transform method, where we show the influence of the incident ion angle, flux, and substrate surface temperature. It is shown that large local geometry variations resulting in quadratic non-linearities in the evolution equation dominate pattern selection and stability at long time scales.

Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period

In this paper, we study bifurcation of the damped Kuramoto-Sivashinsky equation on an odd periodic interval of period $2\lambda$. We fix the control parameter $\alpha \in (0,1)$ and study how the equation bifurcates to attractors as $\lambda$ varies. Using the center manifold analysis, we prove that the bifurcated attractors are homeomorphic to $S^1$ and consist of four or eight singular points and their connecting orbits. We verify the structure of the bifurcated attractors by investigating the stability of each singular point.

Dynamical bifurcation of the damped Kuramoto–Sivashinsky equation

In this paper, we study the primary instability of the damped Kuramoto–Sivashinsky equation under a periodic boundary condition. We prove that it bifurcates from the trivial solution to an attractor which determines the long time dynamics of the system. Using the attractor bifurcation theorem and the center manifold theory, we describe the bifurcated attractor in detail.

Spatio-temporal dynamics of an active, polar, viscoelastic ring.

Constitutive equations for a one-dimensional, active, polar, viscoelastic liquid are derived by treating the strain field as a slow hydrodynamic variable. Taking into account the couplings between strain and polarity allowed by symmetry, the hydrodynamics of an active, polar, viscoelastic body include an evolution equation for the polarity field that generalizes the damped Kuramoto-Sivashinsky equation. Beyond thresholds of the active coupling coefficients between the polarity and the stress or the strain rate, bifurcations of the homogeneous state lead first to stationary waves, then to propagating waves of the strain, stress and polarity fields. I argue that these results are relevant to living matter, and may explain rotating actomyosin rings in cells and mechanical waves in epithelial cell monolayers.

Nanostructure formation of Cu/Si(100) thin film induced by ion beam bombardment

A simple method for fabricating self-organized Cu nano-dots on Si(100) substrate by low energy Ar+ ion beam bombardment of a Cu thin film at room temperature over a large area is demonstrated. The morphological evolution has been investigated using scanning electron microscopy and atomic force microscopy. It was found that nano-ripple patterns formed on a Cu grain surface on a 110 nm thick polycrystalline Cu thin film under normal ion incidence. Uniformly distributed Cu nano-dots were obtained by bombardment of 55 nm thick nano-crystalline Cu thin films. The formation mechanism of the Cu nanostructures was discussed with the aid of numerical simulations using a modified damped Kuramoto–Sivashinsky equation.

Symmetry Reduction of Two-Dimensional Damped Kuramoto—Sivashinsky Equation

In this paper, the problem of determining the largest possible set of symmetries for an important nonlinear dynamical system: the two-dimensional damped Kuramoto—Sivashinsky ((2D) DKS) equation is studied. By applying the basic Lie symmetry method for the (2D) DKS equation, the classical Lie point symmetry operators are obtained. Also, the optimal system of one-dimensional subalgebras of the equation is constructed. The Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. The nonclassical symmetries of the (2D) DKS equation are also investigated.

Numerical simulation of asymptotic states of the damped Kuramoto-Sivashinsky equation

The damped Kuramoto-Sivashinsky equation has emerged as a fundamental tool for the understanding of the onset and evolution of secondary instabilities in a wide range of physical phenomena. Most existing studies about this equation deal with its asymptotic states on one-dimensional settings or on periodic square domains. We utilize a large-scale numerical simulation to investigate the asymptotic states of the damped Kuramoto-Sivashinsky equation on annular two-dimensional geometries and three-dimensional domains. To this end, we propose an accurate, efficient, and robust algorithm based on a recently introduced numerical methodology, namely, isogeometric analysis. We compared our two-dimensional results with several experiments of directed percolation on square and annular geometries, and found qualitative agreement.

Minimization of topological defects in ion-induced ripple patterns on silicon

The evolution of self-organized nanoscale ripple patterns induced by low-energy ion sputtering of silicon is investigated. The quality of the patterns is monitored by calculating a normalized density of topological defects from atomic force microscopy images. A strong dependence of the normalized defect density on the applied ion fluence is observed with a well-pronounced minimum at intermediate fluences. Simulations using the damped Kuramoto–Sivashinsky equation yield good agreement with the experiments and are further used to study the dynamics of single pattern defects.

Chaotic saddles at the onset of intermittent spatiotemporal chaos

In a recent study [Rempel and Chian, Phys. Rev. Lett. 98, 014101 (2007)], it has been shown that nonattracting chaotic sets (chaotic saddles) are responsible for intermittency in the regularized long-wave equation that undergoes a transition to spatiotemporal chaos (STC) via quasiperiodicity and temporal chaos. In the present paper, it is demonstrated that a similar mechanism is present in the damped Kuramoto-Sivashinsky equation. Prior to the onset of STC, a spatiotemporally chaotic saddle coexists with a spatially regular attractor. After the transition to STC, the chaotic saddle merges with the attractor, generating intermittent bursts of STC that dominate the post-transition dynamics.

Surface structuring by multiple ion beams

We study surface patterns produced by low energy ion sputtering of amorphous targets based on a generalized version of the damped Kuramoto-Sivashinsky equation with the focus on the simultaneous incidence of four identical ion beams. This setup allows us to quantitatively investigate the relevance of anisotropic ion-induced effective surface diffusion in theory and in experiments. We find that hexagonal, square, and superstructure patterns can develop depending on the strength of that anisotropy. Squares and superstructures serve as clear indicators for the presence of ion-induced effective surface diffusion. We also discuss the possibility of simulating multiple ion beam incidence by stepwise rotation of the target.

Continuum modeling of sputter erosion under normal incidence: Interplay between nonlocality and nonlinearity

Under specific experimental circumstances, sputter erosion on semiconductor materials exhibits highly ordered hexagonal dot-like nanostructures. In a recent attempt to theoretically understand this pattern forming process, Facsko et al. [Phys. Rev. B 69, 153412 (2004)] suggested a nonlocal, damped Kuramoto-Sivashinsky equation as a potential candidate for an adequate continuum model of this self-organizing process. In this study we theoretically investigate this proposal by (i) formally deriving such a nonlocal equation as minimal model from balance considerations, (ii) showing that it can be exactly mapped to a local, damped Kuramoto-Sivashinsky equation, and (iii) inspecting the consequences of the resulting non-stationary erosion dynamics.

Dissipative continuum model for self-organized pattern formation during ion-beam erosion

A continuum model, based on the damped Kuramoto-Sivashinsky equation, is shown to reproduce the morphology evolution during ion sputtering quite successfully. In a very narrow range of the damping parameter $\ensuremath{\alpha},$ the alignment of the structures into hexagonal domains is obtained under normal incidence of ions with striking resemblance to the experimentally observed dot patterns. The origin of this damping factor is discussed.

Marginal and weakly nonlinear stability in spatially developing flows

This work is devoted to revealing the essence of near-critical phenomena in nonlinear problems with nonparallel effects. As a generalization of the well-known concept of linear stability in Fourier space for a parallel basic state, we introduce a new concept valid for nonparallel flows as well. The new picture allows one to demonstrate the possible singular limit to the parallel case. Also, on its basis we derive a weakly nonlinear model valid near criticality. The damped Kuramoto-Sivashinsky equation with variable coefficients is used to illustrate the application of the theory.

Adiabatic hypercooling of binary melts

A binary melt is hypercooled when it is cooled to a temperature below its solidus. In the isothermal limit planar solidification fronts propagate at a constant velocity determined by the kinetic undercooling and are subject to a long-wavelength morphological instability if speeds fall below a critical value. Here we examine the adiabatic limit where the accumulation of a small latent heat release causes the velocity of the interface to slowly decrease through its critical value. The evolution of the hypercooled interface is governed by a damped Kuramoto-Sivashinsky (dKS) equation with coefficients that vary as the interface decelerates. Using this equation we show that morphological transitions are delayed by an amount that reflects both the time the system spends in a stable state and the magnitude of the damping. For a sufficiently large latent heat of fusion the long-wavelength morphological instability is annihilated. Finally, the adiabatic dKS equation predicts late-stage coarsening of the microstructure with length scales that increase as t(1/2). In finite systems this coarsening removes the morphological instability.

Block Spins for Partial Differential Equations

We investigate the use of renormalisation group methods to solve partial differential equations (PDEs) numerically. Our approach focuses on coarse-graining the underlying continuum process as opposed to the conventional numerical analysis method of sampling it. We calculate exactly the coarse-grained or `perfect' Laplacian operator and investigate the numerical effectiveness of the technique on a series of 1+1-dimensional PDEs with varying levels of smoothness in the dynamics: the diffusion equation, the time-dependent Ginzburg-Landau equation, the Swift-Hohenberg equation and the damped Kuramoto-Sivashinsky equation. We find that the renormalisation group is superior to conventional sampling-based discretisations in representing faithfully the dynamics with a large grid spacing, introducing no detectable lattice artifacts as long as there is a natural ultra-violet cut off in the problem. We discuss limitations and open problems of this approach.