C. William Gear

The gap-tooth method in particle simulations

We explore the gap-tooth method for multiscale modeling of systems represented by microscopic physics-based simulators, when coarse-grained evolution equations are not available in closed form. A biased random walk particle simulation, motivated by the viscous Burgers equation, serves as an example. We construct macro-to-micro (lifting) and micro-to-macro (restriction) operators, and drive the coarse time-evolution by particle simulations in appropriately coupled microdomains (“teeth”) separated by large spatial gaps. A macroscopically interpolative mechanism for communication between the teeth at the particle level is introduced. The results demonstrate the feasibility of a “closure-on-demand” approach to solving some hydrodynamics problems.

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 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

Analysis of the accuracy and convergence of equation-free projection to a slow manifold

In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris, SIAM J. Appl. Dyn. Syst. 4 (2005) 711–732], we developed a class of iterative algorithms within the context of equation-free methods to approximate low-dimensional, attracting, slow manifolds in systems of differential equations with multiple time scales. For user-specified values of a finite number of the observables, the mth member of the class of algorithms (

Deciding the Nature of the Coarse Equation through Microscopic Simulations: The Baby-Bathwater Scheme

m = 0, 1, \ldots

Intrinsic map dynamics exploration for uncharted effective free-energy landscapes

) finds iteratively an approximation of the appropriate zero of the (m+1)st time derivative of the remaining variables and uses this root to approximate the location of the point on the slow manifold corresponding to these values of the observables. This article is the first of two articles in which the accuracy and convergence of the iterative algorithms are analyzed. Here, we work directly with fast-slow systems, in which there is an explicit small parameter,

Data-Driven Reduction for a Class of Multiscale Fast-Slow Stochastic Dynamical Systems

\varepsilon

Analysis and Computation of a Discrete KdV-Burgers Type Equation with Fast Dispersion and Slow Diffusion

, measuring the separation of time scales. We show that, for each

Equationfree Modeling For Complex Systems

m = 0, 1, \ldots

Coarse-Grained Descriptions of Dynamics for Networks with Both Intrinsic and Structural Heterogeneities

, the fixed point of the iterative algorithm approximates the slow manifold up to and including terms of

Equation-free: The computer-aided analysis of complex multiscale systems

{\mathcal O}(\varepsilon^m)