52,618 research outputs found
Uncertainty Quantification for Linear Hyperbolic Equations with Stochastic Process or Random Field Coefficients
In this paper hyperbolic partial differential equations with random
coefficients are discussed. Such random partial differential equations appear
for instance in traffic flow problems as well as in many physical processes in
random media. Two types of models are presented: The first has a time-dependent
coefficient modeled by the Ornstein--Uhlenbeck process. The second has a random
field coefficient with a given covariance in space. For the former a formula
for the exact solution in terms of moments is derived. In both cases stable
numerical schemes are introduced to solve these random partial differential
equations. Simulation results including convergence studies conclude the
theoretical findings
Monte Carlo evaluation of FADE approach to anomalous kinetics
In this paper we propose a comparison between the CTRW (Monte Carlo) and
Fractional Derivative approaches to the modelling of anomalous diffusion
phenomena in the presence of an advection field. Galilei variant and invariant
schemes are revised.Comment: 13 pages, 6 figure
Probabilistic analysis of the upwind scheme for transport
We provide a probabilistic analysis of the upwind scheme for
multi-dimensional transport equations. We associate a Markov chain with the
numerical scheme and then obtain a backward representation formula of
Kolmogorov type for the numerical solution. We then understand that the error
induced by the scheme is governed by the fluctuations of the Markov chain
around the characteristics of the flow. We show, in various situations, that
the fluctuations are of diffusive type. As a by-product, we prove that the
scheme is of order 1/2 for an initial datum in BV and of order 1/2-a, for all
a>0, for a Lipschitz continuous initial datum. Our analysis provides a new
interpretation of the numerical diffusion phenomenon
Stochastic representation of the Reynolds transport theorem: revisiting large-scale modeling
We explore the potential of a formulation of the Navier-Stokes equations
incorporating a random description of the small-scale velocity component. This
model, established from a version of the Reynolds transport theorem adapted to
a stochastic representation of the flow, gives rise to a large-scale
description of the flow dynamics in which emerges an anisotropic subgrid
tensor, reminiscent to the Reynolds stress tensor, together with a drift
correction due to an inhomogeneous turbulence. The corresponding subgrid model,
which depends on the small scales velocity variance, generalizes the Boussinesq
eddy viscosity assumption. However, it is not anymore obtained from an analogy
with molecular dissipation but ensues rigorously from the random modeling of
the flow. This principle allows us to propose several subgrid models defined
directly on the resolved flow component. We assess and compare numerically
those models on a standard Green-Taylor vortex flow at Reynolds 1600. The
numerical simulations, carried out with an accurate divergence-free scheme,
outperform classical large-eddies formulations and provides a simple
demonstration of the pertinence of the proposed large-scale modeling
Modelling of radionuclide migration through the geosphere with radial basis function method and geostatistics
The modelling of radionuclide transport through the geosphere is necessary in the safety assessment of repositories for radioactive waste. A number of key geosphere processes need to be considered when predicting the movement of radionuclides through the geosphere. The most important input data are obtained from field measurements, which are not available for all regions of interest. For example, the hydraulic conductivity, as input parameter, varies from place to place. In such cases geostatistical science offers a variety of spatial estimation procedures. To assess the a long term safety of a radioactive waste disposal system, mathematical models are used to describe the complicated groundwater flow, chemistry and potential radionuclide migration through geological formations. The numerical solution of partial differential equations (PDEs) has usually been obtained by finite difference methods (FDM), finite element methods (FEM), or finite volume methods (FVM). Kansa introduced the concept of solving PDEs using radial basis functions (RBFs) for hyperbolic, parabolic and elliptic PDEs. The aim of this study was to present a relatively new approach to the modelling of radionuclide migration through the geosphere using radial basis functions methods and to determine the average and sample variance of radionuclide concentration with regard to spatial variability of hydraulic conductivity modelled by a geostatistical approach. We will also explore residual errors and their influence on optimal shape parameters
Crossover from the chiral to the standard universality classes in the conductance of a quantum wire with random hopping only
The conductance of a quantum wire with off-diagonal disorder that preserves a
sublattice symmetry (the random hopping problem with chiral symmetry) is
considered. Transport at the band center is anomalous relative to the standard
problem of Anderson localization both in the diffusive and localized regimes.
In the diffusive regime, there is no weak-localization correction to the
conductance and universal conductance fluctuations are twice as large as in the
standard cases. Exponential localization occurs only for an even number of
transmission channels in which case the localization length does not depend on
whether time-reversal and spin rotation symmetry are present or not. For an odd
number of channels the conductance decays algebraically. Upon moving away from
the band center transport characteristics undergo a crossover to those of the
standard universality classes of Anderson localization. This crossover is
calculated in the diffusive regime.Comment: 22 pages, 9 figure
Uncertainty quantification for kinetic models in socio-economic and life sciences
Kinetic equations play a major rule in modeling large systems of interacting
particles. Recently the legacy of classical kinetic theory found novel
applications in socio-economic and life sciences, where processes characterized
by large groups of agents exhibit spontaneous emergence of social structures.
Well-known examples are the formation of clusters in opinion dynamics, the
appearance of inequalities in wealth distributions, flocking and milling
behaviors in swarming models, synchronization phenomena in biological systems
and lane formation in pedestrian traffic. The construction of kinetic models
describing the above processes, however, has to face the difficulty of the lack
of fundamental principles since physical forces are replaced by empirical
social forces. These empirical forces are typically constructed with the aim to
reproduce qualitatively the observed system behaviors, like the emergence of
social structures, and are at best known in terms of statistical information of
the modeling parameters. For this reason the presence of random inputs
characterizing the parameters uncertainty should be considered as an essential
feature in the modeling process. In this survey we introduce several examples
of such kinetic models, that are mathematically described by nonlinear Vlasov
and Fokker--Planck equations, and present different numerical approaches for
uncertainty quantification which preserve the main features of the kinetic
solution.Comment: To appear in "Uncertainty Quantification for Hyperbolic and Kinetic
Equations
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