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

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- by Yingda Cheng, Irene M. Gamba and Jennifer Proft PDF
- Math. Comp.
**81**(2012), 153-190 Request permission

## 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.## References

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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
- MR Author ID: 811395
- Email: ycheng@math.utexas.edu
**Irene M. Gamba**- Affiliation: Department of Mathematics and ICES, University of Texas at Austin, Austin, Texas 78712
- MR Author ID: 241132
- Email: gamba@math.utexas.edu
**Jennifer Proft**- Affiliation: ICES, University of Texas at Austin, Austin, Texas 78712
- Email: jennifer@ices.utexas.edu
- Received by editor(s): July 23, 2010
- Received by editor(s) in revised form: October 27, 2010
- Published electronically: June 15, 2011
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**81**(2012), 153-190 - MSC (2010): Primary 65M60, 76P05, 74S05
- DOI: https://doi.org/10.1090/S0025-5718-2011-02504-4
- MathSciNet review: 2833491