Journal of Scientific Computing

, Volume 79, Issue 2, pp 969–991 | Cite as

Filtered Hyperbolic Moment Method for the Vlasov Equation

  • Yana Di
  • Yuwei Fan
  • Zhenzhong Kou
  • Ruo Li
  • Yanli WangEmail author


In this paper, we investigate the effect of the filter for the hyperbolic moment equations (HME) (Cai et al. in Commun Pure Appl Math 67(3):464–518, 2014; Cai et al. in SIAM J Sci Comput 35(6):A2807–A2831, 2013) of the Vlasov–Poisson equations and propose a novel quasi time-consistent filter to suppress the numerical recurrence effect. By taking properties of HME into consideration, the filter preserves a lot of physical properties of HME, including Galilean invariance and conservation of mass, momentum and energy. We present two viewpoints—collisional viewpoint and dissipative viewpoint—to dissect the filter, and show that the filtered hyperbolic moment method can be treated as a solver of the Vlasov equation. Numerical simulations of the linear Landau damping and two stream instability demonstrate the effectiveness of the filter in restraining recurrence arising from particle streaming. Both the analysis and the numerical results indicate that the filtered method can capture the evolution of the Vlasov equation, even when phase mixing and filamentation dominate.


Hyperbolic moment equations Vlasov equation Filter Landau damping Two-stream instability 



This research of Y. Di is supported in part by the Natural Science Foundation of China (Grant Nos. 11771437 and 91630208). And that of Y. Wang is supported in part by the Natural Science Foundation of China No. 11501042. R. Li is supported in part by the National Natural Science Foundation of China (Grant No. 9163030002).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Yana Di
    • 1
    • 2
  • Yuwei Fan
    • 3
  • Zhenzhong Kou
    • 4
  • Ruo Li
    • 5
  • Yanli Wang
    • 6
    Email author
  1. 1.LSEC, NCMIS, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.Department of MathematicsStanford UniversityStanfordUSA
  4. 4.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  5. 5.CAPT, LMAM and School of Mathematical SciencesPeking UniversityBeijingChina
  6. 6.College of EngineeringPeking UniversityBeijingChina

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