Enhanced Convergence Estimates for Semi-Lagrangian Schemes Application to the Vlasov--Poisson Equation

Charles, Frédérique; Després, Bruno; Mehrenberger, Michel

doi:10.1137/110851511

Cited by 28 publications

(28 citation statements)

References 17 publications

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“…-For the coarse mesh, we observe oscillations that increase when the time step get smaller; this is due to the fact that we use cubic spline interpolation which behave bad when the time step becomes small (see [10,24]). …”

Section: Kelvin-helmholtz Instability In a Periodic Box With Colella mentioning

confidence: 97%

Guiding center simulations on curvilinear grids

Hamiaz¹,

Mehrenberger²,

Back³

et al. 2016

ESAIM: Proc.

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Abstract. Semi-Lagrangian guiding center simulations are performed on sinusoidal perturbations of cartesian grids, and on deformed polar grid with different boundary conditions. Key ingredients are: the use of a B-spline finite element solver for the Poisson equation and the classical backward semiLagrangian method (BSL) for the advection. We are able to reproduce standard Kelvin-Helmholtz and diocotron instability tests on such grids. When the perturbation leads to a strong distorted mesh, we observe that the solution differs if one takes standard numerical parameters that are used in the cartesian reference case. We can recover good results together with correct mass conservation, by diminishing the time step.Résumé. Des simulations semi-lagrangiennes du modèle centre-guide sont effectuées sur des perturbations sinusoïdales de maillages cartésiens et sur des maillages polaires déformés. Les ingrédients clés sont: l'utilisation d'un solveuréléments finis splines pour l'équation de Poisson et la méthode semi-Lagrangienne en arrière classique (BSL) pour l'advection. Nous pouvons reproduire des tests standards d'instabilité de Kelvin-Helmholtz et diocotron sur ces grilles. Lorsque la déformation du maillage est forte, on observe que la solution diffère, si on prend les paramètres numériques standards qui sont utilisés dans le cas cartésien de référence. On peut retrouver néanmoins les bons résultats et une conservation de masse correcte, en diminuant le pas de temps.

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Section: Kelvin-helmholtz Instability In a Periodic Box With Colella mentioning

confidence: 97%

Guiding center simulations on curvilinear grids

Hamiaz¹,

Mehrenberger²,

Back³

et al. 2016

ESAIM: Proc.

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“…To deal with the one-dimensional advection equations, a semi-Lagrangian method is used (see [9,6,7]). Since the characteristics can be solved exactly in this case (a does not depend on z), the error produced by the scheme comes from the splitting (error in time) and from the interpolation step (error in x and v).…”

Section: Numerical Examplesmentioning

confidence: 99%

“…all the L p norms of f ). Note that in this work, we do not address the delicate question of phase space approximation and focus only on time discretization effects (see [2,6,20]). …”

Section: Introductionmentioning

confidence: 99%

High-order Hamiltonian splitting for the Vlasov–Poisson equations

et al. 2016

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Abstract. We consider the Vlasov-Poisson equation in a Hamiltonian framework and derive new time splitting methods based on the decomposition of the Hamiltonian functional between the kinetic and electric energy. Assuming smoothness of the solutions, we study the order conditions of such methods. It appears that these conditions are of Runge-Kutta-Nyström type. In the one dimensional case, the order conditions can be further simplified, and efficient methods of order 6 with a reduced number of stages can be constructed. In the general case, high-order methods can also be constructed using explicit computations of commutators. Numerical results are performed and show the benefit of using high-order splitting schemes in that context. Complete and self-contained proofs of convergence results and rigorous error estimates are also given.

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“…when the diagonal part of the tensor ε 0 ω (r) vanishes and the non-diagonal part remains non-zero. As shown in [9,10], for a 1D-counterpart of (3), in this case the time-harmonic electric field componentÊ x is not necessarily square integrable (for a demonstration of this behaviour with the help of a simpler example, namely the Budden problem, see e.g. [10]).…”

Section: Introductionmentioning

confidence: 98%

A numerical study of the solution of X-Mode equations around the hybrid resonance

et al. 2016

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Abstract. The hybrid resonance is a physical phenomenon that appears for example in the heating of plasma, and as such is of scientific interest for the development of the ITER project. In this paper we focus on solutions of low regularity to Maxwell equations in magnetized plasmas. Our main purpose is three-fold. First, we aim at investigating the finite element approximation of the frequency-domain problem. Second, we would like to study the resonant solutions in the time domain, with the help of two different finite difference approximations. We finally compare the results with the ones obtained in the frequency domain, by numerical examination of the limiting absorption and limited amplitude principles.Résumé. La résonnance hybride est un phénomène physique qui apparait par exemple lorsque l'on chauffe un plasma. Il est en particulier d'intérêt scientifique dans le cadre du développement du projet ITER. Dans ce papier, nous nous concentrons sur des solutions faiblement régulières deséquations de Maxwell pour les plasmas magnétiques. Notre but est ici triple. Dans un premier temps, nous approchons numériquement la formulation fréquentielleà l'aide d'éléments finis. Dans un deuxième temps, nousétudions les solutions résonnantes dans le domaine temporel,à l'aide de deux méthodes distinctes de différences finies. Finalement, nous comparons numériquement les solutions fréquentielles avec le comportement en temps long des solutions temporelles, dans le cadre des principes d'amplitude et d'absorption limite.

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Enhanced Convergence Estimates for Semi-Lagrangian Schemes Application to the Vlasov--Poisson Equation

Cited by 28 publications

References 17 publications

Guiding center simulations on curvilinear grids

Guiding center simulations on curvilinear grids

High-order Hamiltonian splitting for the Vlasov–Poisson equations

A numerical study of the solution of X-Mode equations around the hybrid resonance

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