Hamiltonian particle-in-cell methods for Vlasov-Maxwell equations

Yang, He; Sun, Yajuan; Qin, Hong; Liu, Jian

doi:10.1063/1.4962573

Cited by 59 publications

(92 citation statements)

References 41 publications

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“…Charge conservative geometric electromagnetic PIC methods in cubic meshes are developed by Xiao et al [33], and it has been applied to ideal two-fluid systems as well [97]. FEEC is also used to construct geometric PIC algorithms with discrete gauge symmetry and charged conservation law [35,36]. The charge conservation property can exist in non-Hamiltonian discrete particle-field systems [11, 13, 22, 101-103, 128, 129].…”

Section: Structure-preserving Geometric Relativistic Symplectic Pic Smentioning

confidence: 99%

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Structure-preserving geometric particle-in-cell methods for Vlasov-Maxwell systems

Xiao¹,

Qin²,

Liu³

2018

Plasma Sci. Technol.

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Recent development of structure-preserving geometric particle-in-cell (PIC) algorithms for Vlasov-Maxwell systems is summarized. With the arriving of 100 petaflop and exaflop computing power, it is now possible to carry out direct simulations of multi-scale plasma dynamics based on first-principles. However, standard algorithms currently adopted by the plasma physics community do not possess the long-term accuracy and fidelity required in these large-scale simulations. This is because conventional simulation algorithms are based on numerically solving the underpinning differential (or integro-differential) equations, and the algorithms used in general do not preserve the geometric and physical structures of the systems, such as the local energy-momentum conservation law, the symplectic structure, and the gauge symmetry. As a consequence, numerical errors accumulate coherently with time and long-term simulation results are not reliable. To overcome this difficulty and to hardness the power of exascale computers, a new generation of structure-preserving geometric PIC algorithms have been developed. This new generation of algorithms utilizes modern mathematical techniques, such as discrete manifolds, interpolating differential forms, and non-canonical symplectic integrators, to ensure gauge symmetry, space-time symmetry and the conservation of charge, energy-momentum, and the symplectic structure. These highly desired properties are difficult to achieve using the conventional PIC algorithms. In addition to summarizing the recent development and demonstrating practical implementations, several new results are also presented, including a structure-preserving geometric relativistic PIC algorithm, the proof of the correspondence between discrete gauge symmetry and discrete charge conservation law, and a reformulation of the explicit non-canonical symplectic algorithm for the discrete Poisson bracket using the variational approach. Numerical examples are given to verify the advantages of the structure-preserving geometric PIC algorithms in comparison with the conventional PIC methods. * hongqin@ustc.edu.cn i.e., J div d * I,J J J,l + Dt (ρ I,l ) = 0 ,where div d * is the transpose of −∇ d defined in Eq. (A11).If Eq. (149) is satisfied initially, according to Eq. (148),we can see that Eq. (149) will be automatically satisfied for all time steps and there is no need to solve it.Similarly, we can prove that the discrete Lagrangians of the particle-field systems defined by Eqs. (159) and (167) also admit discrete charge conservation law due to the fact that they are gauge symmetric. Numeric examplesUsing the C programming language and the Message Passing Interface (MPI), we have implemented the 2nd-order explicit Hamiltonian splitting PIC (EHSPIC) algorithm, the 1st-order variational symplectic charge-conservative relativistic PIC (VSCRPIC) algorithm, and the conventional Boris-Yee PIC (BYPIC) scheme for comparison study. Among these PIC schemes, the VSCRPIC algorithm is relativistic, and both EHSPIC and VSCRPIC algorithms a...

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Section: Structure-preserving Geometric Relativistic Symplectic Pic Smentioning

confidence: 99%

“…To overcome this serious difficulty and to hardness the power of exascale computers, a new generation of structure-preserving geometric PIC algorithms [30][31][32][33][34][35][36][37][38] have been developed.…”

Section: Introductionmentioning

confidence: 99%

Structure-preserving geometric particle-in-cell methods for Vlasov-Maxwell systems

Xiao¹,

Qin²,

Liu³

2018

Plasma Sci. Technol.

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“…First, by using a field discretization as in Refs. 4,6, and 7, gauge invariance and charge conservation will be preserved exactly. Second, by using the macroparticle formalism from Refs.…”

Section: Introductionmentioning

confidence: 97%

Finite-dimensional collisionless kinetic theory

Burby

2017

Physics of Plasmas

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A collisionless kinetic plasma model may often be cast as an infinite-dimensional noncanonical Hamiltonian system. I show that, when this is the case, the model can be discretized in space and particles while preserving its Hamiltonian structure, thereby producing a finite-dimensional Hamiltonian system that approximates the original kinetic model. I apply the general theory to two example systems: the relativistic Vlasov-Maxwell system with spin, and a gyrokinetic Vlasov-Maxwell system.

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“…If Coulomb collisions are neglected altogether, recent work on Vlasov-Poisson, Vlasov-Maxwell and related systems has provided algorithms that satisfy energy conservation and also preserve other invariants present in the system, such as the momentum and charge conservation, and the divergence-free nature of the magnetic field. For algorithms in the particle-in-cell framework see for example [4][5][6][7][8][9][10][11][12][13][14][15][16], for discontinuous Galerkin methods see [17][18][19][20][21][22][23][24][25] and for other grid-based methods see [26][27][28][29]. Realistic kinetic simulations of plasmas, expanding to macroscopic time scales, however, require also the effects of Coulomb collisions.…”

Section: Introductionmentioning

confidence: 99%

Metriplectic integrators for the Landau collision operator

Kraus

Hirvijoki

2017

Physics of Plasmas

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We present a novel framework for addressing the nonlinear Landau collision integral in terms of finite element and other subspace projection methods. We employ the underlying metriplectic structure of the Landau collision integral and, using a Galerkin discretization for the velocity space, we transform the infinitedimensional system into a finite-dimensional, time-continuous metriplectic system. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of energy, momentum, and particle densities, as well as the production of entropy is demonstrated algebraically for the fully discrete system. Due to the generality of our approach, the conservation properties and the monotonic behavior of entropy are guaranteed for finite element discretizations in general, independently of the mesh configuration.

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Hamiltonian particle-in-cell methods for Vlasov-Maxwell equations

Cited by 59 publications

References 41 publications

Structure-preserving geometric particle-in-cell methods for Vlasov-Maxwell systems

Structure-preserving geometric particle-in-cell methods for Vlasov-Maxwell systems

Finite-dimensional collisionless kinetic theory

Metriplectic integrators for the Landau collision operator

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