Solving the Vlasov–Maxwell equations using Hamiltonian splitting

Li, Yingzhe; Yang, He; Sun, Yajuan; Niesen, Jitse; Qin, Hong; Liu, Jian

doi:10.1016/j.jcp.2019.06.070

Cited by 17 publications

(23 citation statements)

References 39 publications

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“…Once the semi-discretization is performed, the resulting Poisson system has to be integrated in time. Here, a Hamiltonian splitting method (Crouseilles et al 2015;Li et al 2019) is adopted, in which the solution is obtained as compositions of exact solutions of Hamiltonian subsystems. Hence, such resulting schemes are Poisson integrators in the sense of Hairer, Lubich & Wanner (2002).…”

Section: Hamiltonian Splitting Methodsmentioning

confidence: 99%

“…2015; Li et al. 2019) is adopted, in which the solution is obtained as compositions of exact solutions of Hamiltonian subsystems. Hence, such resulting schemes are Poisson integrators in the sense of Hairer, Lubich & Wanner (2002).…”

Section: Hamiltonian Splitting Methodsmentioning

confidence: 99%

“…Subsequently, one has to implement a time discretization for this large ODE system. This is performed by decomposing the discrete Hamiltonian into several subsystems (Crouseilles, Einkemmer & Faou 2015;He et al 2015;Li et al 2019) and using a time-splitting method. It turns out that each of the subsystem can be solved exactly in time, which means that their composition is still a Poisson map and high-order splitting schemes can thus be easily constructed.…”

Section: Introductionmentioning

confidence: 99%

“…2015; Li et al. 2019) and using a time-splitting method. It turns out that each of the subsystem can be solved exactly in time, which means that their composition is still a Poisson map and high-order splitting schemes can thus be easily constructed.…”

Section: Introductionmentioning

confidence: 99%

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Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

Crouseilles

Hervieux

et al. 2021

J. Plasma Phys.

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We propose a numerical scheme to solve the semiclassical Vlasov–Maxwell equations for electrons with spin. The electron gas is described by a distribution function $f(t,{\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ that evolves in an extended 9-dimensional phase space $({\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ , where $\boldsymbol s$ represents the spin vector. Using suitable approximations and symmetries, the extended phase space can be reduced to five dimensions: $(x,{{p_x}}, {\boldsymbol s})$ . It can be shown that the spin Vlasov–Maxwell equations enjoy a Hamiltonian structure that motivates the use of the recently developed geometric particle-in-cell (PIC) methods. Here, the geometric PIC approach is generalized to the case of electrons with spin. Total energy conservation is very well satisfied, with a relative error below $0.05\,\%$ . As a relevant example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, where the electrons are partially or fully spin polarized. It is shown that the Raman instability is very effective in destroying the electron polarization.

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Section: Hamiltonian Splitting Methodsmentioning

confidence: 99%

Section: Hamiltonian Splitting Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

Crouseilles

Hervieux

et al. 2021

J. Plasma Phys.

Self Cite

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“…By preserving the geometric properties inherited by the concerned system, geometric integration methods usually provide superior long time behavior in comparison with traditional numerical methods, and are thus more suitable for large-scale, longterm simulations. In the context of VM system, several structure preserving methods have recently emerged, such as grid based methods [12,26], PIC methods [25,30,38], in which Hamiltonian splitting methods [16,31,37] play an important role.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulations of one laser-plasma model based on Poisson structure

Sun

Crouseilles

2020

Journal of Computational Physics

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In this paper, a bracket structure is proposed for the laser-plasma interaction model introduced in [19], and it is proved by direct calculations that the bracket is Poisson which satisfies the Jacobi identity. Then splitting methods in time are proposed based on the Poisson structure. For the quasirelativistic case, the Hamiltonian splitting leads to three subsystems which can be solved exactly. The conservative splitting is proposed for the fully relativistic case, and three one-dimensional conservative subsystems are obtained. Combined with the splittings in time, in phase space discretization we use the Fourier spectral and finite volume methods. It is proved that the discrete charge and discrete Poisson equation are conserved by our numerical schemes. Numerically, some numerical experiments are conducted to verify good conservations for the charge, energy and Poisson equation.

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Semiexplicit K‐symplectic‐like methods with energy conservation for noncanonical Hamiltonian systems

Zhu,

2024

Numerical Methods Partial

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For the nonseparable noncanonical Hamiltonian systems, we propose efficient K‐symplectic‐like methods which are semiexplicit and energy‐preserving. By introducing two copies of the phase space and constructing an augmented Hamiltonian, we can separate the noncanonical Hamiltonian system into two explicitly integrable parts. Subsequently, explicit K‐symplectic methods can be constructed by using the splitting and composing method. To enforce constraints on the two copies of the phase space, we provide two transformations with energy conservation property. This enables us to obtain semiexplicit K‐symplectic‐like methods that preserve energy. Two algorithms are provided to implement the semiexplicit K‐symplectic‐like methods with energy conservation and their convergence has been proved. Numerical results on two noncanonical Hamiltonian systems demonstrate that the energy errors of our proposed methods remain bounded within machine precision over long time without exhibiting energy drift. Furthermore, the proposed methods exhibit superior computational efficiency compared to the canonicalized symplectic methods of the same order.

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Solving the Vlasov–Maxwell equations using Hamiltonian splitting

Cited by 17 publications

References 39 publications

Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

Numerical simulations of one laser-plasma model based on Poisson structure

Semiexplicit K‐symplectic‐like methods with energy conservation for noncanonical Hamiltonian systems

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