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Journal of Scientific Computing

, Volume 62, Issue 2, pp 555–574 | Cite as

Asymptotic-Preserving Exponential Methods for the Quantum Boltzmann Equation with High-Order Accuracy

  • Jingwei Hu
  • Qin LiEmail author
  • Lorenzo Pareschi
Article

Abstract

In this paper we develop high order asymptotic preserving methods for the spatially inhomogeneous quantum Boltzmann equation. We follow the work in Li and Pareschi (J Comput Phys 259:402–420, 2014) where asymptotic preserving exponential Runge–Kutta methods for the classical inhomogeneous Boltzmann equation were constructed. A major difficulty here is related to the non Gaussian steady states characterizing the quantum kinetic behavior. We show that the proposed schemes achieve high-order accuracy uniformly in time for all Planck constants ranging from classical regime to quantum regime, and all Knudsen number ranging from kinetic regime to fluid regime. Computational results are presented for both Bose gas and Fermi gas.

Keywords

Quantum Boltzmann equation Asymptotic preserving methods Exponential Runge–Kutta schemes 

Mathematics Subject Classification

65L04 65L06 35Q20 82C10 

Notes

Acknowledgments

We would like to express our gratitude to the NSF Grant RNMS11-07444 (KI-Net), and CSCAMM, University of Maryland for holding the conference “Quantum Systems: A Mathematical Journey from Few to Many Particles” in May 2013, during which this work was initiated.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute for Computational Engineering and Sciences (ICES) and Bureau of Economic Geology (BEG)The University of Texas at AustinAustinUSA
  2. 2.Department of Computing + Mathematical Sciences (CMS), The Annenberg CenterCalifornia Institute of TechnologyPasadenaUSA
  3. 3.Department of Mathematics and Computer ScienceUniversity of FerraraFerraraItaly

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