Analysis of an Asymptotic Preserving Scheme for the Euler–Poisson System in the Quasineutral Limit

Degond, Pierre; Liu, Jian Guo; Vignal, Marie-Hélène

doi:10.1137/070690584

Cited by 40 publications

(41 citation statements)

References 26 publications

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“…The reformulated Poisson equation has been previously proposed in the framework of fluid models in [10,12,14], and in [2,13] for plasma kinetic models.…”

Section: Reformulation Of the Poisson Equationmentioning

confidence: 99%

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Asymptotic-Preserving Particle-In-Cell method for the Vlasov–Poisson system near quasineutrality

Degond

Deluzet

Navoret

et al. 2010

Journal of Computational Physics

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This paper deals with the numerical resolution of the Vlasov-Poisson system in a nearly quasineutral regime by Particle-In-Cell (PIC) methods. In this regime, classical PIC methods are subject to stability constraints on the time and space steps related to the small Debye length and large plasma frequency. Here, we propose an "Asymptotic-Preserving" PIC scheme which is not subject to these limitations. Additionally, when the plasma period and Debye length are small compared to the time and space steps, this method provides a consistent PIC discretization of the quasineutral Vlasov equation. We perform several one-dimensional numerical experiments which provide a solid validation of the method and its underlying concepts, and compare the method with classical PIC and Direct-Implicit methods.

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“…The reformulated Poisson equation has been previously proposed in the framework of fluid models in [10,12,14], and in [2,13] for plasma kinetic models.…”

Section: Reformulation Of the Poisson Equationmentioning

confidence: 99%

“…Previous works on AP methods for the quasineutral limit have been devoted to the Euler-Poisson problem [10,12,14] and to Eulerian schemes for the Vlasov-Poisson problem [2]. The quasineutral limit in plasmas has been theoretically investigated in [5,11,20,21,24,37].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic-Preserving Particle-In-Cell method for the Vlasov–Poisson system near quasineutrality

Degond

Deluzet

Navoret

et al. 2010

Journal of Computational Physics

Self Cite

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show abstract

“…In [21], it has been remarked that a system of this kind, i.e. semi-discretized in time, from the stability point of view gives analogous results of a full discretization where central numerical derivatives are employed for the space derivatives.…”

Section: Linear Stability Analysis Of Scheme (25) In the Fluid Limitmentioning

confidence: 99%

Multiscale Schemes for the BGK--Vlasov--Poisson System in the Quasi-Neutral and Fluid Limits. Stability Analysis and First Order Schemes

Crouseilles¹,

Dimarco²,

Vignal³

2016

Multiscale Model. Simul.

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This paper deals with the development and the analysis of asymptotic stable and consistent schemes in the joint quasi-neutral and fluid limits for the collisional Vlasov-Poisson system. In these limits, the classical explicit schemes suffer from time step restrictions due to the small plasma period and Knudsen number. To solve this problem, we propose a new scheme stable for choices of time steps independent from the small scales dynamics and with comparable computational cost with respect to standard explicit schemes. In addition, this scheme reduces automatically to consistent discretizations of the underlying asymptotic systems. In this first work on this subject, we propose a first order in time scheme and we perform a relative linear stability analysis to deal with such problems. The framework we propose permits to extend this approach to high order schemes in the next future. We finally show the capability of the method in dealing with small scales through numerical experiments.

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“…If the right spatial discretization is chosen, this system can be solved efficiently for ρ n+1 using Fast Fourier Transform techniques. An important feature of this Helmholtz equation is that it is uniformly elliptic for any ε [6,9]. The updated momentum (ρ n+1 u n+1 ) is then obtained from the momentum equation (10).…”

Section: Time Discretization Of the Split Systemsmentioning

confidence: 99%

An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

Haack

Jin

Liu

2012

Commun. comput. phys.

114

124

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The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods, and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system, and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.

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Analysis of an Asymptotic Preserving Scheme for the Euler–Poisson System in the Quasineutral Limit

Cited by 40 publications

References 26 publications

Asymptotic-Preserving Particle-In-Cell method for the Vlasov–Poisson system near quasineutrality

Asymptotic-Preserving Particle-In-Cell method for the Vlasov–Poisson system near quasineutrality

Multiscale Schemes for the BGK--Vlasov--Poisson System in the Quasi-Neutral and Fluid Limits. Stability Analysis and First Order Schemes

An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

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