In this article we present an a posteriori error estimator for the
spatial-stochastic error of a Galerkin-type discretisation of an
initial value problem for a random hyperbolic conservation law. For
the stochastic discretisation we use the Stochastic Galerkin method
and for the spatial-temporal discretisation of the Stochastic Galerkin
system a Runge-Kutta Discontinuous Galerkin method. The estimator
is obtained using smooth reconstructions of the discrete solution.
Combined with the relative entropy stability framework of Dafermos
dafermos2005hyperbolic, this leads to computable error bounds
for the space-stochastic discretisation error. \\ Moreover, it turns
out that the error estimator admits a splitting into one part representing
the spatial error, and a remaining term, which can be interpreted
as the stochastic error. This decomposition allows us to balance
the errors arising from spatial and stochastic discretisation. We
conclude with some numerical examples confirming the theoretical
findings.
%0 Report
%1 giesselmann2017posteriori
%A Giesselmann, J.
%A Meyer, F.
%A Rohde, C.
%D 2017
%K imported vorlaeufig
%T A posteriori error analysis for random scalar conservation laws using
the Stochastic Galerkin method.
%U https://arxiv.org/abs/1709.04351
%V (submitted)
%X In this article we present an a posteriori error estimator for the
spatial-stochastic error of a Galerkin-type discretisation of an
initial value problem for a random hyperbolic conservation law. For
the stochastic discretisation we use the Stochastic Galerkin method
and for the spatial-temporal discretisation of the Stochastic Galerkin
system a Runge-Kutta Discontinuous Galerkin method. The estimator
is obtained using smooth reconstructions of the discrete solution.
Combined with the relative entropy stability framework of Dafermos
dafermos2005hyperbolic, this leads to computable error bounds
for the space-stochastic discretisation error. \\ Moreover, it turns
out that the error estimator admits a splitting into one part representing
the spatial error, and a remaining term, which can be interpreted
as the stochastic error. This decomposition allows us to balance
the errors arising from spatial and stochastic discretisation. We
conclude with some numerical examples confirming the theoretical
findings.
@techreport{giesselmann2017posteriori,
abstract = {In this article we present an a posteriori error estimator for the
spatial-stochastic error of a Galerkin-type discretisation of an
initial value problem for a random hyperbolic conservation law. For
the stochastic discretisation we use the Stochastic Galerkin method
and for the spatial-temporal discretisation of the Stochastic Galerkin
system a Runge-Kutta Discontinuous Galerkin method. The estimator
is obtained using smooth reconstructions of the discrete solution.
Combined with the relative entropy stability framework of Dafermos
\cite{dafermos2005hyperbolic}, this leads to computable error bounds
for the space-stochastic discretisation error. \\ Moreover, it turns
out that the error estimator admits a splitting into one part representing
the spatial error, and a remaining term, which can be interpreted
as the stochastic error. This decomposition allows us to balance
the errors arising from spatial and stochastic discretisation. We
conclude with some numerical examples confirming the theoretical
findings. },
added-at = {2018-07-20T10:54:15.000+0200},
author = {Giesselmann, J. and Meyer, F. and Rohde, C.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2b425eb1a23f331bac52e9f306c34e6d6/mhartmann},
interhash = {d62dec65886d21b9d7177beb7f24a692},
intrahash = {b425eb1a23f331bac52e9f306c34e6d6},
keywords = {imported vorlaeufig},
owner = {meyerfn},
timestamp = {2018-07-20T08:54:15.000+0200},
title = {A posteriori error analysis for random scalar conservation laws using
the Stochastic Galerkin method.},
url = {https://arxiv.org/abs/1709.04351},
volume = {(submitted)},
year = 2017
}