arXiv subjects

I. G. Kevrekidis

Publications and source records attributed to I. G. Kevrekidis.

At least 55 records · Page 3Linked to original sources

arXiv2005

The Moment Map: Nonlinear dynamics of density evolution via a few moments

We explore situations in which certain stochastic and high-dimensional deterministic systems behave effectively as low-dimensional dynamical systems. We define and study moment maps, maps on spaces of low-order moments of evolving distributions, as a means of understanding equations-free multiscale algorithms for these systems. We demonstrate how nonlinearity arises in these maps and how this results in the stabilization of metastable states. Examples are shown for a hierarchy of models, ranging from simple stochastic differential equations to molecular dynamics simulations of a particle in contact with a heat bath.

cond-mat.other↗

arXiv2005

Exploring Hydrodynamic Closures for the Lid-driven Micro-cavity

A minimal kinetic model is used to study analytically and numerically flows at a micrometer scale. Using the lid-driven microcavity as an illustrative example, the interplay between kinetics and hydrodynamics is quantitatively visualized. The validity of various theories of non-equilibrium thermodynamics of flowing systems is tested in this nontrivial microflow.

cond-mat.stat-mech↗

arXiv2005

Equation-free dynamic renormalization in a glassy compaction model

Combining dynamic renormalization with equation-free computational tools, we study the apparently self-similar evolution of void distribution dynamics in the diffusion-deposition problem proposed by Stinchcombe and Depken [Phys. Rev. Lett. 88, 125701 (2002)]. We illustrate fixed point and dynamic approaches, forward as well as backward in time.

cond-mat.soft↗

arXiv2005

Vortices in Bose-Einstein Condensates: Some Recent Developments

In this brief review we summarize a number of recent developments in the study of vortices in Bose-Einstein condensates, a topic of considerable theoretical and experimental interest in the past few years. We examine the generation of vortices by means of phase imprinting, as well as via dynamical instabilities. Their stability is subsequently examined in the presence of purely magnetic trapping, and in the combined presence of magnetic and optical trapping. We then study pairs of vortices and their interactions, illustrating a reduced description in terms of ordinary differential equations for the vortex centers. In the realm of two vortices we also consider the existence of stable dipole clusters for two-component condensates. Last but not least, we discuss mesoscopic patterns formed by vortices, the so-called vortex lattices and analyze some of their intriguing dynamical features. A number of interesting future directions are highlighted.

cond-mat.other↗

arXiv2004

An Equation-Free Approach to Nonlinear Control: Coarse Feedback Linearization With Pole-Placement

We present an application of equation-free computation to the coarse-grained feedback linearization problem of nonlinear systems described by microscopic/stochastic simulators. Feedback linearization with pole placement requires the solution of a functional equation involving the macroscopic (coarse-grained) system model. In the absence of such a closed-form model, short, appropriately initialized bursts of microscopic simulation are designed and performed, and their results used to estimate on demand the quantities required for the numerical solution of the (explicitly unavailable) functional equation. Our illustrative example is a kinetic Monte Carlo realization of a simplified heterogeneous catalytic reaction scheme

nlin.CG↗

arXiv2004

Higher order accuracy in the gap-tooth scheme for large-scale solutions using microscopic simulators

We are developing a framework for multiscale computation which enables models at a ``microscopic'' level of description, for example Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators, to perform modelling tasks at the ``macroscopic'' length scales of interest. The plan is to use the microscopic rules restricted to small patches of the domain, the ``teeth'', followed by interpolation to estimate macroscopic fields in the ``gaps''. The challenge addressed here is to find general boundary conditions for the patches of microscopic simulators that appropriately connect the widely separated ``teeth'' to achieve high order accuracy over the macroscale. Here we start exploring the issues in the simplest case when the microscopic simulator is the quintessential example of a partial differential equation. For this case analytic solutions provide comparisons. We argue that classic high-order interpolation provides patch boundary conditions which achieve arbitrarily high-order consistency in the gap-tooth scheme, and with care are numerically stable. The high-order consistency is demonstrated on a class of linear partial differential equations in two ways: firstly using the dynamical systems approach of holistic discretisation; and secondly through the eigenvalues of selected numerical problems. When applied to patches of microscopic simulations these patch boundary conditions should achieve efficient macroscale simulation.

math.DS↗

arXiv2004

Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes

The long-term dynamics of many dynamical systems evolve on an attracting, invariant "slow manifold" that can be parameterized by a few observable variables. Yet a simulation using the full model of the problem requires initial values for all variables. Given a set of values for the observables parameterizing the slow manifold, one needs a procedure for finding the additional values such that the state is close to the slow manifold to some desired accuracy. We consider problems whose solution has a singular perturbation expansion, although we do not know what it is nor have any way to compute it. We show in this paper that, under some conditions, computing the values of the remaining variables so that their (m+1)-st time derivatives are zero provides an estimate of the unknown variables that is an mth-order approximation to a point on the slow manifold in sense to be defined. We then show how this criterion can be applied approximately when the system is defined by a legacy code rather than directly through closed form equations.

physics.comp-ph↗

arXiv2004

Newton-Krylov solvers for time-steppers

We study how the Newton-GMRES iteration can enable dynamic simulators (time-steppers) to perform fixed-point and path-following computations.For a class of dissipative problems, whose dynamics are characterized by a slow manifold, the Jacobian matrices in such computations are compact perturbations of the identity. We examine the number of GMRES iterations required for each nonlinear iteration as a function of the dimension of the slow subspace and the time-stepper reporting horizon. In a path-following computation, only a small number (one or two) of additional GMRES iterations is required.

math.DS↗

arXiv2003

From molecular dynamics to coarse self-similar solutions: A simple example using equation-free computation

In the context of the recently developed "equation-free" approach to the computer-assisted analysis of complex systems, we illustrate the computation of coarsely self-similar solutions. Dynamic renormalization and fixed point algorithms for the macroscopic density dynamics are applied to the results of short bursts of appropriately initialized molecular dynamics in a simple diffusion simulation. The approach holds promise for locating coarse self-similar solutions and the corresponding exponents in a variety of multiscale computational contexts.

physics.comp-ph↗

arXiv2003

Constraint-defined Manifolds: a Legacy Code Approach to Low-dimensional Computation

If the dynamics of an evolutionary differential equation system possess a low-dimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximating the slow manifold, however, may be computationally as challenging as the original problem. If the system is defined by a legacy simulation code or a microscopic simulator, it may be impossible to perform the manipulations needed to directly approximate the slow manifold. In this paper we demonstrate that with the knowledge only of a set of ``slow'' variables that can be used to {\it parameterize} the slow manifold, we can conveniently compute, using a legacy simulator, on a nearby manifold. Forward and reverse integration, as well as the location of fixed points are illustrated for a discretization of the Chafee-Infante PDE for parameter values for which an Inertial Manifold is known to exist, and can be used to validate the computational results.

physics.comp-ph↗

arXiv2003

The Stirred Tank Reactor Polymer Electrolyte Membrane Fuel Cell

The design and operation of a differential Polymer Electrolyte Membrane (PEM) fuel cell is described. The fuel cell design is based on coupled Stirred Tank Reactors (STR); the gas phase in each reactor compartment was well mixed. The characteristic times for reactant flow, gas phase diffusion and reaction were chosen so that the gas compositions at both the anode and cathode are uniform. The STR PEM fuel cell is one-dimensional; the only spatial gradients are transverse to the membrane. The STR PEM fuel cell was employed to examine fuel cell start- up, and its dynamic responses to changes in load, temperature and reactant flow rates. Multiple time scales in systems response are found to correspond to water absorption by the membrane, water transport through the membrane and stress-related mechanical changes of the membrane.

physics.chem-ph↗

arXiv2003

Application of Coarse Integration to Bacterial Chemotaxis

We have developed and implemented a numerical evolution scheme for a class of stochastic problems in which the temporal evolution occurs on widely-separated time scales, and for which the slow evolution can be described in terms of a small number of moments of an underlying probability distribution. We demonstrate this method via a numerical simulation of chemotaxis in a population of motile, independent bacteria swimming in a prescribed gradient of a chemoattractant. The microscopic stochastic model, which is simulated using a Monte Carlo method, uses a simplified deterministic model for excitation/adaptation in signal transduction, coupled to a realistic, stochastic description of the flagellar motor. We show that projective time integration of ``coarse'' variables can be carried out on time scales long compared to that of the microscopic dynamics. Our coarse description is based on the spatial cell density distribution. Thus we are assuming that the system ``closes'' on this variable so that it can be described on long time scales solely by the spatial cell density. Computationally the variables are the components of the density distribution expressed in terms of a few basis functions, given by the singular vectors of the spatial density distribution obtained from a sample Monte Carlo time evolution of the system. We present numerical results and analysis of errors in support of the efficacy of this time-integration scheme.

physics.comp-ph↗

arXiv2003

Effective equations for discrete systems: A time stepper based approach

We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation (Shroff and Keller, 1993). The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Pade approximations.

physics.comp-ph↗

arXiv2003

Coarse Projective kMC Integration: Forward/Reverse Initial and Boundary Value Problems

In "equation-free" multiscale computation a dynamic model is given at a fine, microscopic level; yet we believe that its coarse-grained, macroscopic dynamics can be described by closed equations involving only coarse variables. These variables are typically various low-order moments of the distributions evolved through the microscopic model. We consider the problem of integrating these unavailable equations by acting directly on kinetic Monte Carlo microscopic simulators, thus circumventing their derivation in closed form. In particular, we use projective multi-step integration to solve the coarse initial value problem forward in time as well as backward in time (under certain conditions). Macroscopic trajectories are thus traced back to unstable, source-type, and even sometimes saddle-like stationary points, even though the microscopic simulator only evolves forward in time. We also demonstrate the use of such projective integrators in a shooting boundary value problem formulation for the computation of "coarse limit cycles" of the macroscopic behavior, and the approximation of their stability through estimates of the leading "coarse Floquet multipliers".

nlin.CG↗

arXiv2003

Dark-in-Bright Solitons in Bose-Einstein Condensates with Attractive Interactions

We demonstrate a possibility to generate localized states in effectively one-dimensional Bose-Einstein condensates with a negative scattering length in the form of a dark soliton in the presence of an optical lattice (OL) and/or a parabolic magnetic trap. We connect such structures with twisted localized modes (TLMs) that were previously found in the discrete nonlinear Schr{ö}dinger equation. Families of these structures are found as functions of the OL strength, tightness of the magnetic trap, and chemical potential, and their stability regions are identified. Stable bound states of two TLMs are also found. In the case when the TLMs are unstable, their evolution is investigated by means of direct simulations, demonstrating that they transform into large-amplitude fundamental solitons. An analytical approach is also developed, showing that two or several fundamental solitons, with the phase shift $π$ between adjacent ones, may form stable bound states, with parameters quite close to those of the TLMs revealed by simulations. TLM structures are found numerically and explained analytically also in the case when the OL is absent, the condensate being confined only by the magnetic trap.

cond-mat.soft↗

arXiv2003

On 3+1 Dimensional Friedman-Robertson-Walker Universes with Matter

We examine the dynamical behavior of matter coupled to gravity in the context of a linear Klein-Gordon equation coupled to a Friedman-Robertson-Walker metric. The resulting ordinary differential equations can be decoupled, the effect of gravity being traced in rendering the equation for the scalar field nonlinear. We obtain regular (in the massless case) and asymptotic (in the massive case) solutions for the resulting matter field and discuss their ensuing finite time blowup in the light of earlier findings. Finally, some potentially interesting connections of these blowups with features of focusing in the theory of nonlinear partial differential equations are outlined, suggesting the potential relevance of a nonlinear theory of quantum cosmology.

gr-qc↗

arXiv2003

Feedback Loops Between Fields and Underlying Space Curvature: an Augmented Lagrangian Approach

We demonstrate a systematic implementation of coupling between a scalar field and the geometry of the space (curve, surface, etc.) which carries the field. This naturally gives rise to a feedback mechanism between the field and the geometry. We develop a systematic model for the feedback in a general form, inspired by a specific implementation in the context of molecular dynamics (the so-called Rahman-Parrinello molecular dynamics, or RP-MD). We use a generalized Lagrangian that allows for the coupling of the space's metric tensor (the first fundamental form) to the scalar field, and add terms motivated by RP-MD. We present two implementations of the scheme: one in which the metric is only time-dependent [which gives rise to ordinary differential equation (ODE) for its temporal evolution], and one with spatio-temporal dependence [wherein the metric's evolution is governed by a partial differential equation (PDE)]. Numerical results are reported for the (1+1)-dimensional model with a nonlinearity of the sine-Gordon type.

nlin.PS↗

arXiv2003

Enabling Dynamic Process Simulators to Perform Alternative Tasks: A Time-stepper Based Toolkit for Computer-Aided Analysis

We discuss computational superstructures that, using repeated, appropriately initialized short calls, enable temporal process simulators to perform alternative tasks such as fixed point computation, stability analysis and projective integration. We illustrate these concepts through the acceleration of a gPROMS-based Rapid Pressure Swing Adsorption simulation, and discuss their scope and possible extensions.

math.OC↗