In this paper hyperbolic partial differential equations (PDEs) with
random coefficients are discussed. We consider the challenging problem
of flux functions with coefficients modeled by spatiotemporal random
fields. Those fields are given by correlated Gaussian random fields in
space and Ornstein-Uhlenbeck processes in time. The resulting system of
equations consists of a stochastic differential equation for each random
parameter coupled to the hyperbolic conservation law. We de fine an
appropriate solution concept in this setting and analyze errors and
convergence of discretization methods. A novel discretization framework,
based on Monte Carlo finite volume methods, is presented for the robust
computation of moments of solutions to those random hyperbolic PDEs. We
showcase the approach on two examples which appear in applications-the
magnetic induction equation and linear acoustics both with a
spatiotemporal random background velocity field.
SimTech, University of Stuttgart, 70569 Stuttgart, Germany
(andrea.barth@mathematik.unistuttgart.de). This author's work was
supported by the German Research Foundation (DFG) as part of the Cluster
of Excellence in Simulation Technology (EXC 310/2) at the University of
Stuttgart, and it is gratefully acknowledged.
%0 Journal Article
%1 ISI:000385283400013
%A Barth, Andrea
%A Fuchs, Franz G.
%C 3600 UNIV CITY SCIENCE CENTER, PHILADELPHIA, PA 19104-2688 USA
%D 2016
%I SIAM PUBLICATIONS
%J SIAM JOURNAL ON SCIENTIFIC COMPUTING
%K Carlo Gaussian Monte Ornstein-Uhlenbeck differential equation; field; field} finite flux function; hyperbolic method; partial process; quantification; random spatiotemporal uncertainty volume {stochastic
%N 4
%P A2209-A2231
%R 10.1137/15M1027723
%T UNCERTAINTY QUANTIFICATION FOR HYPERBOLIC CONSERVATION LAWS WITH FLUX
COEFFICIENTS GIVEN BY SPATIOTEMPORAL RANDOM FIELDS
%V 38
%X In this paper hyperbolic partial differential equations (PDEs) with
random coefficients are discussed. We consider the challenging problem
of flux functions with coefficients modeled by spatiotemporal random
fields. Those fields are given by correlated Gaussian random fields in
space and Ornstein-Uhlenbeck processes in time. The resulting system of
equations consists of a stochastic differential equation for each random
parameter coupled to the hyperbolic conservation law. We de fine an
appropriate solution concept in this setting and analyze errors and
convergence of discretization methods. A novel discretization framework,
based on Monte Carlo finite volume methods, is presented for the robust
computation of moments of solutions to those random hyperbolic PDEs. We
showcase the approach on two examples which appear in applications-the
magnetic induction equation and linear acoustics both with a
spatiotemporal random background velocity field.
@article{ISI:000385283400013,
abstract = {{In this paper hyperbolic partial differential equations (PDEs) with
random coefficients are discussed. We consider the challenging problem
of flux functions with coefficients modeled by spatiotemporal random
fields. Those fields are given by correlated Gaussian random fields in
space and Ornstein-Uhlenbeck processes in time. The resulting system of
equations consists of a stochastic differential equation for each random
parameter coupled to the hyperbolic conservation law. We de fine an
appropriate solution concept in this setting and analyze errors and
convergence of discretization methods. A novel discretization framework,
based on Monte Carlo finite volume methods, is presented for the robust
computation of moments of solutions to those random hyperbolic PDEs. We
showcase the approach on two examples which appear in applications-the
magnetic induction equation and linear acoustics both with a
spatiotemporal random background velocity field.}},
added-at = {2017-05-18T11:32:12.000+0200},
address = {{3600 UNIV CITY SCIENCE CENTER, PHILADELPHIA, PA 19104-2688 USA}},
affiliation = {{Barth, A (Reprint Author), Univ Stuttgart, SimTech, D-70569 Stuttgart, Germany.
Barth, Andrea, Univ Stuttgart, SimTech, D-70569 Stuttgart, Germany.
Fuchs, Franz G., SINTEF, N-0314 Oslo, Norway.}},
author = {Barth, Andrea and Fuchs, Franz G.},
author-email = {{andrea.barth@mathematik.uni-stuttgart.de
franzgeorgfuchs@gmail.com}},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2ca15e451be40b14c5bec014bafe54360/hermann},
doi = {{10.1137/15M1027723}},
eissn = {{1095-7197}},
funding-acknowledgement = {{German Research Foundation (DFG) as part of Cluster of Excellence in
Simulation Technology at the University of Stuttgart {[}EXC 310/2]}},
funding-text = {{SimTech, University of Stuttgart, 70569 Stuttgart, Germany
(andrea.barth@mathematik.unistuttgart.de). This author's work was
supported by the German Research Foundation (DFG) as part of the Cluster
of Excellence in Simulation Technology (EXC 310/2) at the University of
Stuttgart, and it is gratefully acknowledged.}},
interhash = {b1b958721ff8d51a5d30f7154c6f3414},
intrahash = {ca15e451be40b14c5bec014bafe54360},
issn = {{1064-8275}},
journal = {{SIAM JOURNAL ON SCIENTIFIC COMPUTING}},
keywords = {Carlo Gaussian Monte Ornstein-Uhlenbeck differential equation; field; field} finite flux function; hyperbolic method; partial process; quantification; random spatiotemporal uncertainty volume {stochastic},
keywords-plus = {{FINITE-VOLUME METHODS; LINEAR TRANSPORT-EQUATION;
DIFFERENTIAL-EQUATIONS; ADVECTION EQUATION; POLYNOMIAL CHAOS; SCHEMES;
MULTIDIMENSIONS; SPEED}},
language = {{English}},
number = {{4}},
number-of-cited-references = {{46}},
pages = {{A2209-A2231}},
publisher = {{SIAM PUBLICATIONS}},
research-areas = {{Mathematics}},
times-cited = {{0}},
timestamp = {2017-05-18T09:32:12.000+0200},
title = {{UNCERTAINTY QUANTIFICATION FOR HYPERBOLIC CONSERVATION LAWS WITH FLUX
COEFFICIENTS GIVEN BY SPATIOTEMPORAL RANDOM FIELDS}},
type = {{Article}},
volume = {{38}},
web-of-science-categories = {{Mathematics, Applied}},
year = {{2016}}
}