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Filtered Hyperbolic Moment Method for the Vlasov Equation

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. LSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yana Di

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yana Di

  3. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Yuwei Fan

  4. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Zhenzhong Kou

  5. CAPT, LMAM and School of Mathematical Sciences, Peking University, Beijing, China

    Ruo Li

  6. College of Engineering, Peking University, Beijing, China

    Yanli Wang

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  1. Yana Di
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  3. Zhenzhong Kou
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  4. Ruo Li
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  5. Yanli Wang
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Correspondence to Yanli Wang.

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Di, Y., Fan, Y., Kou, Z. et al. Filtered Hyperbolic Moment Method for the Vlasov Equation. J Sci Comput 79, 969–991 (2019). https://doi.org/10.1007/s10915-018-0882-8

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