TU Darmstadt / ULB / TUbiblio

A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method

Meyer, Fabian and Rohde, Christian and Giesselmann, Jan (2019):
A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method.
In: IMA Journal of Numerical Analysis, ISSN 0272-4979,
DOI: 10.1093/imanum/drz004,
[Online-Edition: https://doi.org/10.1093/imanum/drz004],
[Article]

Abstract

In this article we present an a posteriori error estimator for the spatial–stochastic error of a Galerkin-type discretization of an initial value problem for a random hyperbolic conservation law. For the stochastic discretization we use the stochastic Galerkin method and for the spatial–temporal discretization 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 (2016, Hyperbolic Conservation Laws in Continuum Physics, 4th edn., Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences, vol. 325, Berlin, Springer, pp. xxxviii+826), this leads to computable error bounds for the space–stochastic discretization 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 discretization. We conclude with some numerical examples confirming the theoretical findings.

Item Type: Article
Erschienen: 2019
Creators: Meyer, Fabian and Rohde, Christian and Giesselmann, Jan
Title: A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method
Language: English
Abstract:

In this article we present an a posteriori error estimator for the spatial–stochastic error of a Galerkin-type discretization of an initial value problem for a random hyperbolic conservation law. For the stochastic discretization we use the stochastic Galerkin method and for the spatial–temporal discretization 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 (2016, Hyperbolic Conservation Laws in Continuum Physics, 4th edn., Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences, vol. 325, Berlin, Springer, pp. xxxviii+826), this leads to computable error bounds for the space–stochastic discretization 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 discretization. We conclude with some numerical examples confirming the theoretical findings.

Journal or Publication Title: IMA Journal of Numerical Analysis
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Numerical Analysis and Scientific Computing
Date Deposited: 24 Feb 2020 10:06
DOI: 10.1093/imanum/drz004
Official URL: https://doi.org/10.1093/imanum/drz004
Additional Information:

drz004

Export:
Suche nach Titel in: TUfind oder in Google
Send an inquiry Send an inquiry

Options (only for editors)
Show editorial Details Show editorial Details