Asymptotic-Preserving Exponential Methods for the Quantum Boltzmann Equation with High-Order Accuracy
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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.
KeywordsQuantum Boltzmann equation Asymptotic preserving methods Exponential Runge–Kutta schemes
Mathematics Subject Classification65L04 65L06 35Q20 82C10
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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