Ioannis G. Kevrekidis

Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders

Two‐dimensional unsteady flows in complex geometries that are characterized by simple (low‐dimensional) dynamical behavior are considered. Detailed spectral element simulations are performed, and the proper orthogonal decomposition or POD (also called method of empirical eigenfunctions) is applied to the resulting data for two examples: the flow in a periodically grooved channel and the wake of an isolated circular cylinder. Low‐dimensional dynamical models for these systems are obtained using the empirically derived global eigenfunctions in the spectrally discretized Navier–Stokes equations. The short‐ and long‐term accuracy of the models is studied through simulation, continuation, and bifurcation analysis. Their ability to mimic the full simulations for Reynolds numbers (Re) beyond the values used for eigenfunction extraction is evaluated. In the case of the grooved channel, where the primary horizontal wave number of the flow is imposed from the channel periodicity and so remains unchanged with Re, th...

On the computation of inertial manifolds

Abstract A modified Galerkin (the “Euler-Galerkin”) algorithmj for the computational of inertial manifolds is described and applied to reaction diffusion and the Kuramoto-Sivashinsky (KS) equation. In the context of the (KS) equation, a low-dimensional Euler-Galerkin approximation ( n = 3) is distinctly superior to the traditional Galerkin of the same dimension, and comparable to a traditional Galerkin of a much higher dimension ( n = 16).

Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation

A numerical and analytical study of the Kuramoto–Sivashinsky partial differential equation (PDE) in one spatial dimension with periodic boundary conditions is presented. The structure, stability, and bifurcation characteristics of steady state and time-dependent solutions of the PDE for values of the parameter

Coarse molecular dynamics of a peptide fragment: Free energy, kinetics, and long-time dynamics computations

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Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations

less than 40 are examined. The numerically observed primary and secondary bifurcations of steady states, as well as bifurcations to constant speed traveling waves (limit cycles), are analytically verified. Persistent homoclinic and heteroclinic saddle connections are observed and explained via the system symmetries and fixed point subspaces of appropriate isotropy subgroups of

'Coarse' integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods

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Inherent noise can facilitate coherence in collective swarm motion

. Their effect on the system dynamics is discussed, and several tertiary bifurcations, observed numerically, are presented.

Analysis of drag and virtual mass forces in bubbly suspensions using an implicit formulation of the lattice Boltzmann method

We present a “coarse molecular dynamics” approach and apply it to studying the kinetics and thermodynamics of a peptide fragment dissolved in water. Short bursts of appropriately initialized simulations are used to infer the deterministic and stochastic components of the peptide motion parametrized by an appropriate set of coarse variables. Techniques from traditional numerical analysis (Newton–Raphson, coarse projective integration) are thus enabled; these techniques help analyze important features of the free-energy landscape (coarse transition states, eigenvalues and eigenvectors, transition rates, etc.). Reverse integration of coarse variables backward in time can assist escape from free energy minima and trace low-dimensional free energy surfaces. To illustrate the coarse molecular dynamics approach, we combine multiple short (0.5 ps) replica simulations to map the free energy surface of the “alanine dipeptide” in water, and to determine the ∼1/(1000 ps) rate of interconversion between the two stable...

Equation-Free Multiscale Computation: Algorithms and Applications

Abstract We evaluate several alternative methods for the approximation of inertial manifolds for the one-dimensional Kuramoto-Sivashinsky equation (KSE). A method motivated by the dynamics originally developed for the Navier-Stokes equation is adapted for the KSE. Rigorous error estimates are obtained and compared to those of other methods introduced in the literature. Formal relationships between these other methods and the one introduced here are established. Numerical bifurcation diagrams of the various approximate inertial forms for the KSE are presented. We discuss the correspondence between the rigorous error estimates and the accuracy of the computational results. These methods can be adapted to other dissipative partial differential equations.

Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis

Abstract We present a time-stepper based approach to the ‘coarse’ integration and stability/bifurcation analysis of distributed reacting system models. The methods we discuss are applicable to systems for which the traditional modeling approach through macroscopic evolution equations (usually partial differential equations, PDEs) is not possible because the PDEs are not available in closed form. If an alternative, microscopic (e.g. Monte Carlo or Lattice Boltzmann) description of the physics is available, we illustrate how this microscopic simulator can be enabled (through a computational superstructure) to perform certain integration and numerical bifurcation analysis tasks directly at the coarse, systems level. This approach, when successful, can circumvent the derivation of accurate, closed form, macroscopic PDE descriptions of the system. The direct ‘systems level’ analysis of microscopic process models, facilitated through such numerical ‘enabling technologies’, may, if practical, advance our understanding and use of nonequilibrium systems.