DG-IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

Chen, Zheng; Liu, Liu; Mu, Lin

doi:10.1007/s10915-017-0439-2

Title: DG-IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

Abstract

In this paper, we consider the linear transport equation under diffusive scaling and with random inputs. The method is based on the generalized polynomial chaos approach in the stochastic Galerkin framework. Several theoretical aspects will be addressed. Additionally, a uniform numerical stability with respect to the Knudsen number ϵ, and a uniform in ϵ error estimate is given. For temporal and spatial discretizations, we apply the implicit–explicit scheme under the micro–macro decomposition framework and the discontinuous Galerkin method, as proposed in Jang et al. (SIAM J Numer Anal 52:2048–2072, 2014) for deterministic problem. Lastly, we provide a rigorous proof of the stochastic asymptotic-preserving (sAP) property. Extensive numerical experiments that validate the accuracy and sAP of the method are conducted.

Authors:

^[1]; Liu, Liu ^[2]; Mu, Lin ^[1]

Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Division
Univ. of Wisconsin, Madison, WI (United States). Department of Mathematics

Publication Date:: Wed May 03 00:00:00 EDT 2017

Research Org.:: Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1356966

Grant/Contract Number:: AC05-00OR22725

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Scientific Computing

Additional Journal Information:: Journal Volume: 73; Journal Issue: 2-3; Journal ID: ISSN 0885-7474

Publisher:: Springer

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Kinetic transport equation; Uncertainty quantification; Asymptotic preserving; Random inputs; Discontinuous Galerkin; IMEX; Generalized polynomial chaos

Citation Formats


                    Chen, Zheng, Liu, Liu, and Mu, Lin. DG-IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings.  United States: N. p., 2017. 
Web.  doi:10.1007/s10915-017-0439-2.

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                    Chen, Zheng, Liu, Liu, & Mu, Lin. DG-IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings.  United States.  https://doi.org/10.1007/s10915-017-0439-2

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                    Chen, Zheng, Liu, Liu, and Mu, Lin. Wed .  
"DG-IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings".  United States.  https://doi.org/10.1007/s10915-017-0439-2.  https://www.osti.gov/servlets/purl/1356966.

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@article{osti_1356966,

  title        = {DG-IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings},

  author       = {Chen, Zheng and Liu, Liu and Mu, Lin},

  abstractNote = {In this paper, we consider the linear transport equation under diffusive scaling and with random inputs. The method is based on the generalized polynomial chaos approach in the stochastic Galerkin framework. Several theoretical aspects will be addressed. Additionally, a uniform numerical stability with respect to the Knudsen number ϵ, and a uniform in ϵ error estimate is given. For temporal and spatial discretizations, we apply the implicit–explicit scheme under the micro–macro decomposition framework and the discontinuous Galerkin method, as proposed in Jang et al. (SIAM J Numer Anal 52:2048–2072, 2014) for deterministic problem. Lastly, we provide a rigorous proof of the stochastic asymptotic-preserving (sAP) property. Extensive numerical experiments that validate the accuracy and sAP of the method are conducted.},

  doi          = {10.1007/s10915-017-0439-2},

  journal      = {Journal of Scientific Computing},

  number       = 2-3,

  volume       = 73,

  place        = {United States},

  year         = {Wed May 03 00:00:00 EDT 2017},

  month        = {Wed May 03 00:00:00 EDT 2017}

}

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Works referenced in this record:

Diffusion approximation and computation of the critical size
journal, February 1984

Bardos, C.; Santos, R.; Sentis, R.
Transactions of the American Mathematical Society, Vol. 284, Issue 2
DOI: 10.1090/S0002-9947-1984-0743736-0

Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments
journal, May 2011

Zhang, Xiangxiong; Shu, Chi-Wang
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 467, Issue 2134
DOI: 10.1098/rspa.2011.0153

A boundary matching micro/macro decomposition for kinetic equations
journal, April 2011

Lemou, Mohammed; Méhats, Florian
Comptes Rendus Mathematique, Vol. 349, Issue 7-8
DOI: 10.1016/j.crma.2011.02.002

A stochastic Galerkin method for the Boltzmann equation with uncertainty
journal, June 2016

Hu, Jingwei; Jin, Shi
Journal of Computational Physics, Vol. 315
DOI: 10.1016/j.jcp.2016.03.047

Asymptotic solution of neutron transport problems for small mean free paths
journal, January 1974

Larsen, Edward W.; Keller, Joseph B.
Journal of Mathematical Physics, Vol. 15, Issue 1
DOI: 10.1063/1.1666510

Efficient stochastic Galerkin methods for random diffusion equations
journal, February 2009

Xiu, Dongbin; Shen, Jie
Journal of Computational Physics, Vol. 228, Issue 2
DOI: 10.1016/j.jcp.2008.09.008

Methods for hyperbolic systems with stiff relaxation
journal, January 2002

Lowrie, R. B.; Morel, J. E.
International Journal for Numerical Methods in Fluids, Vol. 40, Issue 3-4
DOI: 10.1002/fld.321

Flux-Explicit IMEX Runge--Kutta Schemes for Hyperbolic to Parabolic Relaxation Problems
journal, January 2013

Boscarino, Sebastiano; Russo, Giovanni
SIAM Journal on Numerical Analysis, Vol. 51, Issue 1
DOI: 10.1137/110850803

Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings
journal, May 2015

Jin, Shi; Xiu, Dongbin; Zhu, Xueyu
Journal of Computational Physics, Vol. 289
DOI: 10.1016/j.jcp.2015.02.023

A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit
journal, January 2008

Lemou, Mohammed; Mieussens, Luc
SIAM Journal on Scientific Computing, Vol. 31, Issue 1
DOI: 10.1137/07069479X

On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations
journal, January 2017

Zhang, Xiangxiong
Journal of Computational Physics, Vol. 328
DOI: 10.1016/j.jcp.2016.10.002

Implicit-Explicit Runge--Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit
journal, January 2013

Boscarino, S.; Pareschi, L.; Russo, G.
SIAM Journal on Scientific Computing, Vol. 35, Issue 1
DOI: 10.1137/110842855

Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes
journal, March 2016

Chen, Zheng; Huang, Hongying; Yan, Jue
Journal of Computational Physics, Vol. 308
DOI: 10.1016/j.jcp.2015.12.039

High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments
journal, July 2016

Shu, Chi-Wang
Journal of Computational Physics, Vol. 316
DOI: 10.1016/j.jcp.2016.04.030

High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling
journal, January 2015

Jang, Juhi; Li, Fengyan; Qiu, Jing-Mei
Journal of Computational Physics, Vol. 281
DOI: 10.1016/j.jcp.2014.10.025

Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations
journal, January 1999

Jin, Shi
SIAM Journal on Scientific Computing, Vol. 21, Issue 2
DOI: 10.1137/S1064827598334599

Analysis of Asymptotic Preserving DG-IMEX Schemes for Linear Kinetic Transport Equations in a Diffusive Scaling
journal, January 2014

Jang, Juhi; Li, Fengyan; Qiu, Jing-Mei
SIAM Journal on Numerical Analysis, Vol. 52, Issue 4
DOI: 10.1137/130938955

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Title: DG-IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

Abstract

Citation Formats

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Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments journal, May 2011

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A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit journal, January 2008

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High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments journal, July 2016

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Diffusion approximation and computation of the critical size
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Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments
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A boundary matching micro/macro decomposition for kinetic equations
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A stochastic Galerkin method for the Boltzmann equation with uncertainty
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Asymptotic solution of neutron transport problems for small mean free paths
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