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Two viewpoints on the quasi time-consistent filter show that the filtered hyperbolic moment equations is a solver of the Vlasov equation, and it can predict correct physical phenomena described by Vlasov–Poisson equations

Filtered Hyperbolic Moment Method for the Vlasov Equation

Journal of Scientific Computing, no. 2 (2019): 969.0-991.0

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academic.microsoft.com dblp.uni-trier.de link.springer.com arxiv.org dl.acm.org

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

摘要

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...更多

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简介

The Vlasov equation is the fundamental kinetic equation modeling of the collisionless plasma.
It describes the time evolution of the distribution function f (t, x, v) of a population of charged particles that responds to the self-consistent electromagnetic fields.
The distribution function f (t, x, v) is the number density of the particles at the time t and position x ∈ ⊂ RD with the microscopic velocity v ∈ RD [48].

重点内容

The Vlasov equation is the fundamental kinetic equation modeling of the collisionless plasma
Using the same argument in Sect. 3.2.1, it can be claimed that the filtered hyperbolic moment equations (HME) is a solver of the Vlasov equation and its solution converges to that of the Vlasov equation as M → ∞
To study the convergence of the filtered HME and whether the filter changes the Landau damping rate, we provided two viewpoints: artificial collision operator in Sect. 3.2.1 and artificial dissipation in Sect. 3.2.2 to show that the filtered HME is a solver of the Vlasov equation and can predict the correct Landau damping rate and frequency
We presented a filtered HME for the Vlasov–Poisson equations to suppress the recurrence effects
Due to the careful construction, the filter preserves most of the physical properties of HME, including the conservation of mass, momentum and energy, Galilean invariant and convergence to Vlasov–Poisson equations (VP)
Two viewpoints on the quasi time-consistent filter show that the filtered HME is a solver of the Vlasov equation, and it can predict correct physical phenomena described by VP

方法

Numerical methods for solving the

Vlasov equation have been extensively studied, see for instance [5,24,29,49] and references therein.
With the exponential growth of computing power, the Eulerian numerical methods attract more and more researchers’ attention, and a lot of methods are developed, for example, the continuous finite element methods [49], finite difference methods [14,22], finite volume methods [23], discontinuous Galerkin methods [29,43], the semi-Lagrangian methods [46] and spectral methods [6,10,44]
These methods discretize or approximate the distribution function both in the spatial space and the microscopic velocity space, and they can be used to solve the case that the distribution function has the low-density velocity much more accurately compared to PIC, but may be quite expensive in the high dimension problem

结论

The authors presented a filtered HME for the Vlasov–Poisson equations to suppress the recurrence effects.
The quasi time-consistent property guarantees that the solution to the filtered HME is not sensitive to the time step.
Two viewpoints on the quasi time-consistent filter show that the filtered HME is a solver of the Vlasov equation, and it can predict correct physical phenomena described by VP.
Numerical simulations demonstrate the power of the filter in suppressing recurrences and producing more accurate solutions.

基金

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

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