374 research outputs found
Focusing Revisited: an MN-dynamics Approach
The nonlinear Schr{\"o}dinger (NLS) equation is a ubiquitous example of an
envelope wave equation for conservative, dispersive systems. We revisit here
the problem of self-similar focusing of waves in the case of the focusing NLS
equation through the prism of a dynamic renormalization technique (MN dynamics)
that factors out self-similarity and yields a bifurcation view of the onset of
focusing. As a result, identifying the focusing self-similar solution becomes a
steady state problem. The discretized steady states are subsequently obtained
and their linear stability is numerically examined. The calculations are
performed in the setting of variable index of refraction, in which the onset of
focusing appears as a supercritical bifurcation of a novel type of mixed
Hamiltonian-dissipative dynamical system (reminiscent, to some extent, of a
pitchfork bifurcation).Comment: 6 pages, 2 figure
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.Comment: 4 pages, 4 figures (Minor Modifications; Submitted Version
Computational coarse graining of a randomly forced 1-D Burgers equation
We explore a computational approach to coarse graining the evolution of the
large-scale features of a randomly forced Burgers equation in one spatial
dimension. The long term evolution of the solution energy spectrum appears
self-similar in time. We demonstrate coarse projective integration and coarse
dynamic renormalization as tools that accelerate the extraction of macroscopic
information (integration in time, self-similar shapes, and nontrivial dynamic
exponents) from short bursts of appropriately initialized direct simulation.
These procedures solve numerically an effective evolution equation for the
energy spectrum without ever deriving this equation in closed form.Comment: 21 pages, 7 figure
Coarse-graining the Dynamics of a Driven Interface in the Presence of Mobile Impurities: Effective Description via Diffusion Maps
Developing effective descriptions of the microscopic dynamics of many
physical phenomena can both dramatically enhance their computational
exploration and lead to a more fundamental understanding of the underlying
physics. Previously, an effective description of a driven interface in the
presence of mobile impurities, based on an Ising variant model and a single
empirical coarse variable, was partially successful; yet it underlined the
necessity of selecting additional coarse variables in certain parameter
regimes. In this paper we use a data mining approach to help identify the
coarse variables required. We discuss the implementation of this diffusion map
approach, the selection of a similarity measure between system snapshots
required in the approach, and the correspondence between empirically selected
and automatically detected coarse variables. We conclude by illustrating the
use of the diffusion map variables in assisting the atomistic simulations, and
we discuss the translation of information between fine and coarse descriptions
using lifting and restriction operators.Comment: 28 pages, 10 figure
General tooth boundary conditions for equation free modelling
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
``macroscopic'' length scales of interest. The plan is to use the microscopic
rules restricted to small "patches" of the domain, the "teeth'', using
interpolation to bridge the "gaps". Here we explore general boundary conditions
coupling the widely separated ``teeth'' of the microscopic simulation that
achieve high order accuracy over the macroscale. We present the simplest case
when the microscopic simulator is the quintessential example of a partial
differential equation. We argue that classic high-order interpolation of the
macroscopic field provides the correct forcing in whatever boundary condition
is required by the microsimulator. Such interpolation leads to Tooth Boundary
Conditions which achieve arbitrarily high-order consistency. The high-order
consistency is demonstrated on a class of linear partial differential equations
in two ways: firstly through the eigenvalues of the scheme for selected
numerical problems; and secondly using the dynamical systems approach of
holistic discretisation on a general class of linear \textsc{pde}s. Analytic
modelling shows that, for a wide class of microscopic systems, the subgrid
fields and the effective macroscopic model are largely independent of the tooth
size and the particular tooth boundary conditions. When applied to patches of
microscopic simulations these tooth boundary conditions promise efficient
macroscale simulation. We expect the same approach will also accurately couple
patch simulations in higher spatial dimensions.Comment: 22 page
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