348 research outputs found
Entropy Balance and Dispersive Oscillations in Lattice Boltzmann Models
We conduct an investigation into the dispersive post-shock oscillations in
the entropic lattice-Boltzmann method (ELBM). To this end we use a root finding
algorithm to implement the ELBM which displays fast cubic convergence and
guaranties the proper sign of dissipation. The resulting simulation on the
one-dimensional shock tube shows no benefit in terms of regularization from
using the ELBM over the standard LBGK method. We also conduct an experiment
investigating of the LBGK method using median filtering at a single point per
time step. Here we observe that significant regularization can be achieved.Comment: 18 pages, 4 figures; 13/07/2009 Matlab code added to appendi
Possibility and Impossibility of the Entropy Balance in Lattice Boltzmann Collisions
We demonstrate that in the space of distributions operated on by lattice
Boltzmann methods that there exists a vicinity of the equilibrium where
collisions with entropy balance are possible and, at the same time, there exist
an area of nonequilibrium distributions where such collisions are impossible.
We calculate and graphically represent these areas for some simple entropic
equilibria using single relaxation time models. Therefore it is shown that the
definition of an entropic LBM is incomplete without a strategy to deal with
certain highly nonequilibrium states. Such strategies should be explicitly
stated as they may result in the production of additional entropy.Comment: v2 minor misprint correction
Reciprocal Relations Between Kinetic Curves
We study coupled irreversible processes. For linear or linearized kinetics
with microreversibility, , the kinetic operator is symmetric in
the entropic inner product. This form of Onsager's reciprocal relations implies
that the shift in time, , is also a symmetric operator. This
generates the reciprocity relations between the kinetic curves. For example,
for the Master equation, if we start the process from the th pure state and
measure the probability of the th state (), and,
similarly, measure for the process, which starts at the th pure
state, then the ratio of these two probabilities is constant in
time and coincides with the ratio of the equilibrium probabilities. We study
similar and more general reciprocal relations between the kinetic curves. The
experimental evidence provided as an example is from the reversible water gas
shift reaction over iron oxide catalyst. The experimental data are obtained
using Temporal Analysis of Products (TAP) pulse-response studies. These offer
excellent confirmation within the experimental error.Comment: 6 pages, 1 figure, the final versio
Quasiequilibrium lattice Boltzmann models with tunable bulk viscosity for enhancing stability
Taking advantage of a closed-form generalized Maxwell distribution function [ P. Asinari and I. V. Karlin Phys. Rev. E 79 036703 (2009)] and splitting the relaxation to the equilibrium in two steps, an entropic quasiequilibrium (EQE) kinetic model is proposed for the simulation of low Mach number flows, which enjoys both the H theorem and a free-tunable parameter for controlling the bulk viscosity in such a way as to enhance numerical stability in the incompressible flow limit. Moreover, the proposed model admits a simplification based on a proper expansion in the low Mach number limit (LQE model). The lattice Boltzmann implementation of both the EQE and LQE is as simple as that of the standard lattice Bhatnagar-Gross-Krook (LBGK) method, and practical details are reported. Extensive numerical testing with the lid driven cavity flow in two dimensions is presented in order to verify the enhancement of the stability region. The proposed models achieve the same accuracy as the LBGK method with much rougher meshes, leading to an effective computational speed-up of almost three times for EQE and of more than four times for the LQE. Three-dimensional extension of EQE and LQE is also discussed
Stabilisation of the lattice-Boltzmann method using the Ehrenfests' coarse-graining
The lattice-Boltzmann method (LBM) and its variants have emerged as
promising, computationally efficient and increasingly popular numerical methods
for modelling complex fluid flow. However, it is acknowledged that the method
can demonstrate numerical instabilities, e.g., in the vicinity of shocks. We
propose a simple and novel technique to stabilise the lattice-Boltzmann method
by monitoring the difference between microscopic and macroscopic entropy.
Populations are returned to their equilibrium states if a threshold value is
exceeded. We coin the name Ehrenfests' steps for this procedure in homage to
the vehicle that we use to introduce the procedure, namely, the Ehrenfests'
idea of coarse-graining. The one-dimensional shock tube for a compressible
isothermal fluid is a standard benchmark test for hydrodynamic codes. We
observe that, of all the LBMs considered in the numerical experiment with the
one-dimensional shock tube, only the method which includes Ehrenfests' steps is
capable of suppressing spurious post-shock oscillations.Comment: 4 pages, 9 figure
PCA Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes
Multidimensional data distributions can have complex topologies and variable
local dimensions. To approximate complex data, we propose a new type of
low-dimensional ``principal object'': a principal cubic complex. This complex
is a generalization of linear and non-linear principal manifolds and includes
them as a particular case. To construct such an object, we combine a method of
topological grammars with the minimization of an elastic energy defined for its
embedment into multidimensional data space. The whole complex is presented as a
system of nodes and springs and as a product of one-dimensional continua
(represented by graphs), and the grammars describe how these continua transform
during the process of optimal complex construction. The simplest case of a
topological grammar (``add a node'', ``bisect an edge'') is equivalent to the
construction of ``principal trees'', an object useful in many practical
applications. We demonstrate how it can be applied to the analysis of bacterial
genomes and for visualization of cDNA microarray data using the ``metro map''
representation. The preprint is supplemented by animation: ``How the
topological grammar constructs branching principal components
(AnimatedBranchingPCA.gif)''.Comment: 19 pages, 8 figure
Elastic Maps and Nets for Approximating Principal Manifolds and Their Application to Microarray Data Visualization
Principal manifolds are defined as lines or surfaces passing through ``the
middle'' of data distribution. Linear principal manifolds (Principal Components
Analysis) are routinely used for dimension reduction, noise filtering and data
visualization. Recently, methods for constructing non-linear principal
manifolds were proposed, including our elastic maps approach which is based on
a physical analogy with elastic membranes. We have developed a general
geometric framework for constructing ``principal objects'' of various
dimensions and topologies with the simplest quadratic form of the smoothness
penalty which allows very effective parallel implementations. Our approach is
implemented in three programming languages (C++, Java and Delphi) with two
graphical user interfaces (VidaExpert
http://bioinfo.curie.fr/projects/vidaexpert and ViMiDa
http://bioinfo-out.curie.fr/projects/vimida applications). In this paper we
overview the method of elastic maps and present in detail one of its major
applications: the visualization of microarray data in bioinformatics. We show
that the method of elastic maps outperforms linear PCA in terms of data
approximation, representation of between-point distance structure, preservation
of local point neighborhood and representing point classes in low-dimensional
spaces.Comment: 35 pages 10 figure
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