418 research outputs found
Stability analysis of a hyperbolic stochastic Galerkin formulation for the Aw-Rascle-Zhang model with relaxation
We investigate the propagation of uncertainties in the Aw-Rascle-Zhang model, which belongs to a class of second order traffic flow models described by a system of nonlinear hyperbolic equations. The stochastic quantities are expanded in terms of wavelet-based series expansions. Then, they are projected to obtain a deterministic system for the coefficients in the truncated series. Stochastic Galerkin formulations are presented in conservative form and for smooth solutions also in the corresponding non-conservative form. This allows to obtain stabilization results, when the system is relaxed to a first-order model. Computational tests illustrate the theoretical results
Energy Stable and Structure-Preserving Schemes for the Stochastic Galerkin Shallow Water Equations
The shallow water flow model is widely used to describe water flows in
rivers, lakes, and coastal areas. Accounting for uncertainty in the
corresponding transport-dominated nonlinear PDE models presents theoretical and
numerical challenges that motivate the central advances of this paper. Starting
with a spatially one-dimensional hyperbolicity-preserving,
positivity-preserving stochastic Galerkin formulation of the
parametric/uncertain shallow water equations, we derive an entropy-entropy flux
pair for the system. We exploit this entropy-entropy flux pair to construct
structure-preserving second-order energy conservative, and first- and
second-order energy stable finite volume schemes for the stochastic Galerkin
shallow water system. The performance of the methods is illustrated on several
numerical experiments
Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach
In this paper, we consider Burgers' equation with uncertain boundary and
initial conditions. The polynomial chaos (PC) approach yields a hyperbolic
system of deterministic equations, which can be solved by several numerical
methods. Here, we apply the correction procedure via reconstruction (CPR) using
summation-by-parts operators. We focus especially on stability, which is proven
for CPR methods and the systems arising from the PC approach. Due to the usage
of split-forms, the major challenge is to construct entropy stable numerical
fluxes. For the first time, such numerical fluxes are constructed for all
systems resulting from the PC approach for Burgers' equation. In numerical
tests, we verify our results and show also the advantage of the given ansatz
using CPR methods. Moreover, one of the simulations, i.e. Burgers' equation
equipped with an initial shock, demonstrates quite fascinating observations.
The behaviour of the numerical solutions from several methods (finite volume,
finite difference, CPR) differ significantly from each other. Through careful
investigations, we conclude that the reason for this is the high sensitivity of
the system to varying dissipation. Furthermore, it should be stressed that the
system is not strictly hyperbolic with genuinely nonlinear or linearly
degenerate fields
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
Particle based gPC methods for mean-field models of swarming with uncertainty
In this work we focus on the construction of numerical schemes for the
approximation of stochastic mean--field equations which preserve the
nonnegativity of the solution. The method here developed makes use of a
mean-field Monte Carlo method in the physical variables combined with a
generalized Polynomial Chaos (gPC) expansion in the random space. In contrast
to a direct application of stochastic-Galerkin methods, which are highly
accurate but lead to the loss of positivity, the proposed schemes are capable
to achieve high accuracy in the random space without loosing nonnegativity of
the solution. Several applications of the schemes to mean-field models of
collective behavior are reported.Comment: Communications in Computational Physics, to appea
Structure preserving stochastic Galerkin methods for Fokker-Planck equations with background interactions
This paper is devoted to the construction of structure preserving stochastic
Galerkin schemes for Fokker-Planck type equations with uncertainties and
interacting with an external distribution, that we refer to as a background
distribution. The proposed methods are capable to preserve physical properties
in the approximation of statistical moments of the problem like nonnegativity,
entropy dissipation and asymptotic behaviour of the expected solution. The
introduced methods are second order accurate in the transient regimes and high
order for large times. We present applications of the developed schemes to the
case of fixed and dynamic background distribution for models of collective
behaviour
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