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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations


Authors: Yingda Cheng, Irene M. Gamba and Jennifer Proft
Journal: Math. Comp. 81 (2012), 153-190
MSC (2010): Primary 65M60, 76P05, 74S05
Published electronically: June 15, 2011
MathSciNet review: 2833491
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Abstract: We develop a high-order positivity-preserving discontinuous Galerkin (DG) scheme for linear Vlasov-Boltzmann transport equations (Vlasov-BTE) under the action of quadratically confined electrostatic potentials. The solutions of such BTEs are positive probability distribution functions and it is very challenging to have a mass-conservative, high-order accurate scheme that preserves positivity of the numerical solutions in high dimensions. Our work extends the maximum-principle-satisfying scheme for scalar conservation laws in a recent work by X. Zhang and C.-W. Shu to include the linear Boltzmann collision term. The DG schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy. A discussion of the standard semi-discrete DG schemes for the BTE are included as a foundation for the stability and error estimates for this new scheme. Numerical results of the relaxation models are provided to validate the method.


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Additional Information

Yingda Cheng
Affiliation: Department of Mathematics and ICES, University of Texas at Austin, Austin, Texas 78712
Address at time of publication: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: ycheng@math.utexas.edu

Irene M. Gamba
Affiliation: Department of Mathematics and ICES, University of Texas at Austin, Austin, Texas 78712
Email: gamba@math.utexas.edu

Jennifer Proft
Affiliation: ICES, University of Texas at Austin, Austin, Texas 78712
Email: jennifer@ices.utexas.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2011-02504-4
PII: S 0025-5718(2011)02504-4
Keywords: Boltzmann transport equations, discontinuous Galerkin finite element methods, positivity-preserving schemes, stability, error estimates, relaxation models
Received by editor(s): July 23, 2010
Received by editor(s) in revised form: October 27, 2010
Published electronically: June 15, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.