We describe the structure of an original application of the method of moments to the Vlasov–Poisson system with non constant strong magnetic field in three dimensions of space. Using basis functions which are aligned with the magnetic field, one obtains a Friedrichs system where the kernel of the singular part is made explicit. A projection of the original model on this kernel yields what we call the reduced model. Basic numerical tests of the field illustrate the accuracy of our implementation. A new generating formula for Laguerre polynomials is obtained in the appendix as a byproduct of the analysis.
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@article{CRMECA_2023__351_S1_307_0,
author = {Fr\'ed\'erique Charles and Bruno Despr\'es and Ruiyang Dai and Sever A. Hirstoaga},
title = {Discrete moments models for {Vlasov} equations with non constant strong magnetic limit},
journal = {Comptes Rendus. M\'ecanique},
pages = {307--329},
publisher = {Acad\'emie des sciences, Paris},
volume = {351},
number = {S1},
year = {2023},
doi = {10.5802/crmeca.219},
language = {en},
}
TY - JOUR AU - Frédérique Charles AU - Bruno Després AU - Ruiyang Dai AU - Sever A. Hirstoaga TI - Discrete moments models for Vlasov equations with non constant strong magnetic limit JO - Comptes Rendus. Mécanique PY - 2023 SP - 307 EP - 329 VL - 351 IS - S1 PB - Académie des sciences, Paris DO - 10.5802/crmeca.219 LA - en ID - CRMECA_2023__351_S1_307_0 ER -
%0 Journal Article %A Frédérique Charles %A Bruno Després %A Ruiyang Dai %A Sever A. Hirstoaga %T Discrete moments models for Vlasov equations with non constant strong magnetic limit %J Comptes Rendus. Mécanique %D 2023 %P 307-329 %V 351 %N S1 %I Académie des sciences, Paris %R 10.5802/crmeca.219 %G en %F CRMECA_2023__351_S1_307_0
Frédérique Charles; Bruno Després; Ruiyang Dai; Sever A. Hirstoaga. Discrete moments models for Vlasov equations with non constant strong magnetic limit. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 307-329. doi : 10.5802/crmeca.219. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.219/
[1] Plasma Physics and Controlled Nuclear Fusion, Springer Series on Atomic, Optical, and Plasma Physics, 38, Springer, 2005 | Zbl
[2] Kinetic equations for low frequency instabilities in inhomogeneous plasmas, Phys. Fluids, Volume 23 (1980), pp. 1205-1214 | DOI | MR | Zbl
[3] Foundations of non linear gyrokinetic theory, Rev. Mod. Phys., Volume 79 (2007) no. 2, pp. 421-468 | DOI | Zbl
[4] The drift kinetic equation for toroidal plasmas with large mass velocities, Plasma Phys., Volume 20 (1978) no. 7, pp. 673-678 | DOI
[5] The Vlasov–Poisson system with strong magnetic field, J. Math. Pures Appl., Volume 78 (1999) no. 8, pp. 791-817 | DOI | MR | Zbl
[6] On the Asymptotic Limit of the Three Dimensional Vlasov–Poisson System for Large Magnetic Field: Formal Derivation, J. Stat. Phys., Volume 165 (2016) no. 4, pp. 765-784 | DOI | MR | Zbl
[7] Well-posedness of a diffusive gyro-kinetic model, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011) no. 4, pp. 529-550 | DOI | Numdam | MR | Zbl
[8] Asymptotics of the three dimensional Vlasov equation in the large magnetic field limit, J. Éc. Polytech., Math., Volume 7 (2020), pp. 1009-1067 | DOI | Numdam | MR | Zbl
[9] Massively parallel three dimensional toroidal gyrokinetic flux-tube turbulence simulations, J. Comput. Phys., Volume 161 (2000), pp. 589-604 | Zbl
[10] Linear gyrokinetic simulations using particles for small perpendicular wavelength perturbations, Proc. Joint Varenna-Lausanne International Workshop Theory of Fusion Plasmas (2001), pp. 327-332
[11] Laguerre–Hermite Pseudo-Spectral Velocity Formulation of Gyrokinetics, J. Plasma Phys., Volume 84 (2018) no. 1, 905840108 | DOI
[12] et al. A 5D gyrokinetic full-f global semi-lagrangian code for flux-driven ion turbulence simulations, Comput. Phys. Commun., Volume 207 (2016), pp. 35-68 | DOI | MR | Zbl
[13] A gyrokinetic model for the plasma periphery of tokamak devices, J. Plasma Phys., Volume 86 (2020) no. 2, 905860205 | DOI
[14] Homogenization of the Vlasov equation and of the Vlasov–Poisson system with a strong external magnetic field, Asymptotic Anal., Volume 18 (1998) no. 3-4, pp. 193-213 | MR | Zbl
[15] Long time behavior of the two-dimensional Vlasov equation with a strong external magnetic field, Math. Models Methods Appl. Sci., Volume 10 (2000) no. 4, pp. 539-553 | DOI | MR | Zbl
[16] Gyrokinetics from variational averaging: Existence and error bounds, J. Math. Phys., Volume 59 (2018) no. 8, 082702 | DOI | MR | Zbl
[17] Spectral velocity discretizations for the Vlasov–Maxwell equations, Transp. Theory Stat. Phys., Volume 25 (1996) no. 1, pp. 1-32 | DOI | MR | Zbl
[18] Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form, J. Comput. Phys., Volume 301 (2015), pp. 338-356 | DOI | MR | Zbl
[19] Symmetrization of Vlasov–Poisson equations, SIAM J. Math. Anal., Volume 46 (2014) no. 4, pp. 2554-2580 | DOI | MR | Zbl
[20] A Legendre-Fourier spectral method with exact conservation laws for the Vlasov–Poisson system, J. Comput. Phys., Volume 317 (2016), pp. 82-107 | DOI | MR | Zbl
[21] Vlasov simulations using velocity-scaled hermite representations, J. Comput. Phys., Volume 144 (1998) no. 2, pp. 626-661 | DOI | Zbl
[22] On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov–Poisson system, J. Comput. Phys., Volume 451 (2022), 110881 | MR | Zbl
[23] Conservative discontinuous Galerkin/Hermite Spectral Method for the Vlasov–Poisson System, Commun. Appl. Math. Comput., Volume 4 (2022) no. 1, pp. 34-59 | DOI | MR | Zbl
[24] A Hybrid Discontinuous Galerkin Scheme for Multi-scale Kinetic Equations, J. Comput. Phys., Volume 372 (2018), pp. 841-863 | DOI | MR | Zbl
[25] Modèles cinétiques pour la fusion (2008) (Master notes)
[26] NIST Handbook of Mathematical Functions, Cambridge University Press, 2010 | Zbl
[27] Full-F gyrofluid model, Phys. Plasmas, Volume 20 (2013), 072301 | DOI
[28] A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system, J. Comput. Phys., Volume 317 (2016), pp. 82-107 | DOI | MR | Zbl
[29] Finite element methods for incompressible viscous flow, Numerical methods for fluids (Part 3) (Handbook of Numerical Analysis), Volume 9, North-Holland, 2003, pp. 3-1176 | DOI | Zbl
[30] Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., Volume 7 (1986) no. 3, pp. 856-869 | DOI | MR | Zbl
[31] Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities, Int. J. Numer. Methods Eng., Volume 79 (2009) no. 11, pp. 1309-1331 | DOI | MR | Zbl
[32] Recurrence phenomenon for Vlasov–Poisson simulations on regular finite element mesh, Commun. Comput. Phys., Volume 28 (2020) no. 3, pp. 877-901 | DOI | MR | Zbl
[33] The magnetized vlasov–ampère system and the bernstein–landau paradox, J. Stat. Phys., Volume 183 (2021) no. 2, pp. 1-57 | Zbl
[34] Kinetic models for magnetized plasmas, Ph. D. Thesis, Sorbonne Université, Paris, France (2021)
[35] Waves in a plasma in a magnetic field, Phys. Rev., II Ser., Volume 109 (1958) no. 1, pp. 10-21 | MR | Zbl
[36] Physics of Nonneutral Plasmas, Imperial College Press; World Scientific, 2001 | DOI
[37] Asymptotically stable particle-in-cell methods for the Vlasov–Poisson system with a strong external magnetic field, Mathematics, Volume 54 (2016) no. 2, pp. 1120-1146 | MR | Zbl
[38] Uniformly accurate methods for three dimensional Vlasov equations under strong magnetic field with varying direction, SIAM J. Sci. Comput., Volume 42 (2020) no. 2, p. B520-B547 | DOI | MR | Zbl
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