Finite-dimensional collisionless kinetic theory

Burby, J. W.

doi:10.1063/1.4976849

Cited by 37 publications

(34 citation statements)

References 25 publications

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“…Instead of splines, other Finite Element spaces that form a deRham complex could be used, e.g., mimetic spectral elements or Nédélec elements for one-forms and Raviart-Thomas elements for two-forms. Further, it also should be possible to apply this approach to other systems like the gyrokinetic Vlasov-Maxwell system [16,15], although in this case the necessity for new splitting schemes or other time integration strategies might arise. Energy-preserving time stepping methods might provide an alternative to Hamiltonian splitting algorithms, where a suitable splitting cannot be easily found.…”

Section: Discussionmentioning

confidence: 99%

“…In order to ensure that these properties are also conserved by the fully discrete numerical scheme, the semi-discrete bracket is used in conjunction with Poisson time integrators provided by the previously mentioned splitting method [30,77,42] and higher-order compositions thereof. A semi-discretization of the noncanonical Hamiltonian structure of the relativistic Vlasov-Maxwell system with spin and that for the gyrokinetic Vlasov-Maxwell system have recently been described by Burby [15].…”

Section: Introductionmentioning

confidence: 99%

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GEMPIC: geometric electromagnetic particle-in-cell methods

et al. 2017

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We present a novel framework for finite element particle-in-cell methods based on the discretization of the underlying Hamiltonian structure of the Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains the defining properties of a bracket, anti-symmetry and the Jacobi identity, as well as conservation of its Casimir invariants, implying that the semi-discrete system is still a Hamiltonian system. In order to obtain a fully discrete Poisson integrator, the semi-discrete bracket is used in conjunction with Hamiltonian splitting methods for integration in time. Techniques from finite element exterior calculus ensure conservation of the divergence of the magnetic field and Gauss' law as well as stability of the field solver. The resulting methods are gauge invariant, feature exact charge conservation and show excellent long-time energy and momentum behaviour. Due to the generality of our framework, these conservation properties are guaranteed independently of a particular choice of the finite element basis, as long as the corresponding finite element spaces satisfy certain compatibility conditions.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

GEMPIC: geometric electromagnetic particle-in-cell methods

et al. 2017

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“…In a Vlasov system at a bifurcating inhomogeneous stationary state, the same structure (12) for (generalized) eigenvectors and projection operator can be found in (46) and in (48) respectively (see Appendix A for general spatially onedimensional systems).…”

Section: A Linear Analysismentioning

confidence: 71%

“…First, their Hamiltonian structure is noncanonical, and highly degenerate [2][3][4], which is the origin of the infinite number of conserved quantities, called Casimir invariants. We note this degenerate noncanonical structure has been taken advantage of for instance to design numerical schemes in plasma [9][10][11][12][13] and fluid [14,15] contexts, as well as to derive weakly nonlinear expansions [16]. Second, the linearized Vlasov evolution typically features continuous spectrum on the imaginary axis, and a growing unstable mode can create resonances with part of this spectrum, triggering complex dynamics.…”

Section: Introductionmentioning

confidence: 99%

Towards a classification of bifurcations in Vlasov equations

2020

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“…At least four such Vlasov-Maxwell systems exist and can be used in numerical modeling of plasmas in various branches of science. These are the guiding-center [1], the drift-kinetic [2,3], the gyrokinetic [3,4], and the spin-Vlasov-Maxwell system [5]. They all have a structure similar to equations (1).…”

Section: Introductionmentioning

confidence: 99%

Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems

Hirvijoki¹,

Burby²,

Pfefferlé³

et al. 2020

J. Phys. A: Math. Theor.

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The action principle by Low [Proc. R. Soc. Lond. A 248,[282][283][284][285][286][287] for the classic Vlasov-Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy-and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler-Poincaré formulation of Vlasov-Maxwell-type systems, effectively starting with Low's action and using constrained variations for the Eulerian description of particle motion, has been known for a while [J. Math. Phys., 39,6,, it is hard to come by a documented derivation of the related energy-and momentum-conservation laws in the spirit of the Euler-Poincaré machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guidingcenter Vlasov-Darwin system [Phys. Plasmas 25, 102506]. The present exposition discusses a generic class of local Vlasov-Maxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler-Poincaré framework to its full extent. After reviewing the transition from a Lagrangian picture to an Eulerian one, we demonstrate how symmetries generated by isometries in space lead to conservation laws for linear-and angular-momentum density and how symmetry by time translation produces a conservation law for energy density. We also discuss what happens if no symmetries exist. Finally, two explicit examples will be given -the classic Vlasov-Maxwell and the drift-kinetic Vlasov-Maxwell -and the results expressed in the language of regular vector calculus for familiarity.We start with a slightly modified version of Low's action principle [6]. The purpose of the modification is to introduce the capability to handle a wider class of Vlasov-Maxwell-type 1 By systems related to Vlasov-Maxwell models, we mean 1) genuine Vlasov-Maxwell models that form an infinite-dimensional initial-value problem for the dynamical variables, and 2) the so-called Vlasov-Poisson-Ampère models which provide an initial value problem for the distribution function only and constraint equations for the electromagnetic potentials.

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Finite-dimensional collisionless kinetic theory

Cited by 37 publications

References 25 publications

GEMPIC: geometric electromagnetic particle-in-cell methods

GEMPIC: geometric electromagnetic particle-in-cell methods

Towards a classification of bifurcations in Vlasov equations

Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems

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