68 research outputs found

    Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit

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    This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB) model of plasma physics. This model consists of the pressureless gas dynamics equations coupled with the Poisson equation and where the Boltzmann relation relates the potential to the electron density. If the quasi-neutral assumption is made, the Poisson equation is replaced by the constraint of zero local charge and the model reduces to the Isothermal Compressible Euler (ICE) model. We compare a numerical strategy based on the EPB model to a strategy using a reformulation (called REPB formulation). The REPB scheme captures the quasi-neutral limit more accurately

    High Order Asymptotic Preserving and Classical Semi-implicit RK Schemes for the Euler-Poisson System in the Quasineutral Limit

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    In this paper, the design and analysis of high order accurate IMEX finite volume schemes for the compressible Euler-Poisson (EP) equations in the quasineutral limit is presented. As the quasineutral limit is singular for the governing equations, the time discretisation is tantamount to achieving an accurate numerical method. To this end, the EP system is viewed as a differential algebraic equation system (DAEs) via the method of lines. As a consequence of this vantage point, high order linearly semi-implicit (SI) time discretisation are realised by employing a novel combination of the direct approach used for implicit discretisation of DAEs and, two different classes of IMEX-RK schemes: the additive and the multiplicative. For both the time discretisation strategies, in order to account for rapid plasma oscillations in quasineutral regimes, the nonlinear Euler fluxes are split into two different combinations of stiff and non-stiff components. The high order scheme resulting from the additive approach is designated as a classical scheme while the one generated by the multiplicative approach possesses the asymptotic preserving (AP) property. Time discretisations for the classical and the AP schemes are performed by standard IMEX-RK and SI-IMEX-RK methods, respectively so that the stiff terms are treated implicitly and the non-stiff ones explicitly. In order to discretise in space a Rusanov-type central flux is used for the non-stiff part, and simple central differencing for the stiff part. AP property is also established for the space-time fully-discrete scheme obtained using the multiplicative approach. Results of numerical experiments are presented, which confirm that the high order schemes based on the SI-IMEX-RK time discretisation achieve uniform second order convergence with respect to the Debye length and are AP in the quasineutral limit

    An Asymptotic Preserving and Energy Stable Scheme for the Euler-Poisson System in the Quasineutral Limit

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    An asymptotic preserving and energy stable scheme for the Euler-Poisson system under the quasineutral scaling is designed and analysed. Correction terms are introduced in the convective fluxes and the electrostatic potential, which lead to the dissipation of mechanical energy and the entropy stability. The resolution of the semi-implicit in time finite volume in space fully-discrete scheme involves two steps: the solution of an elliptic problem for the potential and an explicit evaluation for the density and velocity. The proposed scheme possesses several physically relevant attributes, such as the the entropy stability and the consistency with the weak formulation of the continuous Euler-Poisson system. The AP property of the scheme, i.e. the boundedness of the mesh parameters with respect to the Debye length and its consistency with the quasineutral limit system, is shown. The results of numerical case studies are presented to substantiate the robustness and efficiency of the proposed method.Comment: 29 pages, research paper. arXiv admin note: text overlap with arXiv:2206.0606

    An Asymptotic Preserving Scheme for the Euler equations in a strong magnetic field

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    This paper is concerned with the numerical approximation of the isothermal Euler equations for charged particles subject to the Lorentz force. When the magnetic field is large, the so-called drift-fluid approximation is obtained. In this limit, the parallel motion relative to the magnetic field direction splits from perpendicular motion and is given implicitly by the constraint of zero total force along the magnetic field lines. In this paper, we provide a well-posed elliptic equation for the parallel velocity which in turn allows us to construct an Asymptotic-Preserving (AP) scheme for the Euler-Lorentz system. This scheme gives rise to both a consistent approximation of the Euler-Lorentz model when epsilon is finite and a consistent approximation of the drift limit when epsilon tends to 0. Above all, it does not require any constraint on the space and time steps related to the small value of epsilon. Numerical results are presented, which confirm the AP character of the scheme and its Asymptotic Stability

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

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    International audienceThis 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

    Numerical approximation of the Euler-Maxwell model in the quasineutral limit

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    International audienceWe derive and analyze an Asymptotic-Preserving scheme for the Euler-Maxwell system in the quasi-neutral limit. We prove that the linear stability condition on the time-step is independent of the scaled Debye length λ\lambda when λ→0\lambda \to 0. Numerical validation performed on Riemann initial data and for a model Plasma Opening Switch device show that the AP-scheme is convergent to the Euler-Maxwell solution when Δx/λ→0\Delta x/ \lambda \to 0 where Δx\Delta x is the spatial discretization. But, when λ/Δx→0\lambda /\Delta x \to 0, the AP-scheme is consistent with the quasi-neutral Euler-Maxwell system. The scheme is also perfectly consistent with the Gauss equation. The possibility of using large time and space steps leads to several orders of magnitude reductions in computer time and storage

    Degenerate anisotropic elliptic problems and magnetized plasma simulations

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    This paper is devoted to the numerical approximation of a degenerate anisotropic elliptic problem. The numerical method is designed for arbitrary space-dependent anisotropy directions and does not require any specially adapted coordinate system. It is also designed to be equally accurate in the strongly and the mildly anisotropic cases. The method is applied to the Euler-Lorentz system, in the drift-fluid limit. This system provides a model for magnetized plasmas

    Mini-Workshop: Numerics for Kinetic Equations

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