5,001 research outputs found
A three-dimensional multidimensional gas-kinetic scheme for the Navier-Stokes equations under gravitational fields
This paper extends the gas-kinetic scheme for one-dimensional inviscid
shallow water equations (J. Comput. Phys. 178 (2002), pp. 533-562) to
multidimensional gas dynamic equations under gravitational fields. Four
important issues in the construction of a well-balanced scheme for gas dynamic
equations are addressed. First, the inclusion of the gravitational source term
into the flux function is necessary. Second, to achieve second-order accuracy
of a well-balanced scheme, the Chapman-Enskog expansion of the Boltzmann
equation with the inclusion of the external force term is used. Third, to avoid
artificial heating in an isolated system under a gravitational field, the
source term treatment inside each cell has to be evaluated consistently with
the flux evaluation at the cell interface. Fourth, the multidimensional
approach with the inclusion of tangential gradients in two-dimensional and
three-dimensional cases becomes important in order to maintain the accuracy of
the scheme. Many numerical examples are used to validate the above issues,
which include the comparison between the solutions from the current scheme and
the Strang splitting method. The methodology developed in this paper can also
be applied to other systems, such as semi-conductor device simulations under
electric fields.Comment: The name of first author was misspelled as C.T.Tian in the published
paper. 35 pages,9 figure
Diffusion maps embedding and transition matrix analysis of the large-scale flow structure in turbulent Rayleigh--B\'enard convection
By utilizing diffusion maps embedding and transition matrix analysis we
investigate sparse temperature measurement time-series data from
Rayleigh--B\'enard convection experiments in a cylindrical container of aspect
ratio between its diameter () and height (). We consider
the two cases of a cylinder at rest and rotating around its cylinder axis. We
find that the relative amplitude of the large-scale circulation (LSC) and its
orientation inside the container at different points in time are associated to
prominent geometric features in the embedding space spanned by the two dominant
diffusion-maps eigenvectors. From this two-dimensional embedding we can measure
azimuthal drift and diffusion rates, as well as coherence times of the LSC. In
addition, we can distinguish from the data clearly the single roll state (SRS),
when a single roll extends through the whole cell, from the double roll state
(DRS), when two counter-rotating rolls are on top of each other. Based on this
embedding we also build a transition matrix (a discrete transfer operator),
whose eigenvectors and eigenvalues reveal typical time scales for the stability
of the SRS and DRS as well as for the azimuthal drift velocity of the flow
structures inside the cylinder. Thus, the combination of nonlinear dimension
reduction and dynamical systems tools enables to gain insight into turbulent
flows without relying on model assumptions
Local Variational Principle
A generalization of the Gibbs-Bogoliubov-Feynman inequality for spinless
particles is proven and then illustrated for the simple model of a symmetric
double-well quartic potential. The method gives a pointwise lower bound for the
finite-temperature density matrix and it can be systematically improved by the
Trotter composition rule. It is also shown to produce groundstate energies
better than the ones given by the Rayleigh-Ritz principle as applied to the
groundstate eigenfunctions of the reference potentials. Based on this
observation, it is argued that the Local Variational Principle performs better
than the equivalent methods based on the centroid path idea and on the
Gibbs-Bogoliubov-Feynman variational principle, especially in the range of low
temperatures.Comment: 15 pages, 5 figures, one more section adde
Maximum Fidelity
The most fundamental problem in statistics is the inference of an unknown
probability distribution from a finite number of samples. For a specific
observed data set, answers to the following questions would be desirable: (1)
Estimation: Which candidate distribution provides the best fit to the observed
data?, (2) Goodness-of-fit: How concordant is this distribution with the
observed data?, and (3) Uncertainty: How concordant are other candidate
distributions with the observed data? A simple unified approach for univariate
data that addresses these traditionally distinct statistical notions is
presented called "maximum fidelity". Maximum fidelity is a strict frequentist
approach that is fundamentally based on model concordance with the observed
data. The fidelity statistic is a general information measure based on the
coordinate-independent cumulative distribution and critical yet previously
neglected symmetry considerations. An approximation for the null distribution
of the fidelity allows its direct conversion to absolute model concordance (p
value). Fidelity maximization allows identification of the most concordant
model distribution, generating a method for parameter estimation, with
neighboring, less concordant distributions providing the "uncertainty" in this
estimate. Maximum fidelity provides an optimal approach for parameter
estimation (superior to maximum likelihood) and a generally optimal approach
for goodness-of-fit assessment of arbitrary models applied to univariate data.
Extensions to binary data, binned data, multidimensional data, and classical
parametric and nonparametric statistical tests are described. Maximum fidelity
provides a philosophically consistent, robust, and seemingly optimal foundation
for statistical inference. All findings are presented in an elementary way to
be immediately accessible to all researchers utilizing statistical analysis.Comment: 66 pages, 32 figures, 7 tables, submitte
Power-law random walks
We present some new results about the distribution of a random walk whose
independent steps follow a Gaussian distribution with exponent
. In the case we show that a stochastic
representation of the point reached after steps of the walk can be
expressed explicitly for all . In the case we show that the random
walk can be interpreted as a projection of an isotropic random walk, i.e. a
random walk with fixed length steps and uniformly distributed directions.Comment: 5 pages, 4 figure
- …