Comptes Rendus
Short paper
A hierarchy of reduced models to approximate Vlasov–Maxwell equations for slow time variations
Comptes Rendus. Mécanique, Volume 348 (2020) no. 12, pp. 969-981.

We introduce a new family of paraxial asymptotic models that approximate the Vlasov–Maxwell equations in non-relativistic cases. This formulation is nth order accurate in a parameter η, which denotes the ratio between the characteristic velocity of the beam and the speed of light. This family of models is interesting, first because it is simpler than the complete Vlasov–Maxwell equation and then because it allows us to choose the model complexity according to the expected accuracy.

On introduit une nouvelle famille de modèles asymptotiques paraxiaux pour approcher le système d’équations de Vlasov–Maxwell dans le cas non relativiste. Cette formulation est précise à un ordre n (que l’on peut choisir) par rapport à un paramètre η, qui désigne le quotient de la vitesse caractéristique du faisceau par celle de la lumière. L’intéret de cette famille de modèle est, d’une part, qu’elle est plus simple que le système complet des équations de Vlasov–Maxwell, tout en permettant, d’autre part, de choisir le degré de complexité du modèle, en fonction de la précison désirée.

Published online:
DOI: 10.5802/crmeca.50
Keywords: Vlasov–Maxwell equations, Asymptotic analysis, Paraxial model, Reduced models, Non-relativistic
Keywords: Équations de Vlasov–Maxwell, Analyse asymptotique, Modèle paraxial, Modèles réduits, Non relativiste
Franck Assous 1; Yevgeni Furman 1

1 Department of Mathematics, Ariel University, 40700, Ariel, Israel
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Franck Assous; Yevgeni Furman. A hierarchy of reduced models to approximate Vlasov–Maxwell equations for slow time variations. Comptes Rendus. Mécanique, Volume 348 (2020) no. 12, pp. 969-981. doi : 10.5802/crmeca.50.

[1] W. J. Harris U.S. Patent No. 3,271,556, 1966 (Washington, DC: U.S. Patent and Trademark Office)

[2] M. J. Madou Manufacturing Techniques for Microfabrication and Nanotechnology, Vol. 2, CRC Press, 2011

[3] R. B. Miller An Introduction to the Physics of Intense Charged Particle Beams, Springer, 1984

[4] B. Danly; G. Bekefi; R. Davidson; R. Temkin; T. Tran; J. Wurtele Principles of gyrotron powered electromagnetic wigglers for free-electron lasers, IEEE J. Quantum Electron., Volume 23 (1987) no. 1, pp. 103-116 | DOI

[5] T. M. Tran; J. S. Wurtele Free-electron laser simulation techniques, Phys. Rep., Volume 195 (1990) no. 1, pp. 1-21 | DOI

[6] J. D. Lawson The Physics of Charged Particle Beams, Clarendon Press, Oxford, 1988

[7] M. Reiser Theory and Design of Charged Particle Beams, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2008 | DOI

[8] A. Vlasov On the kinetic theory of an assembly of particles with collective interaction, Russ. Phys. J., Volume 9 (1945), pp. 25-40 | MR | Zbl

[9] C. K. Birdsall; A. B. Langdon Plasmas Physics via Computer Simulation, McGraw-Hill, New York, 1985

[10] F. Assous; P. Ciarlet Jr.; S. Labrunie Mathematical Foundations of Computational Electromagnetism, Appl. Math. Sci., AMS, 198, Springer, 2018 | MR | Zbl

[11] F. Assous; P. Degond; E. Heintzé; P. A. Raviart; J. Segré On a finite element method for solving the three dimensional Maxwell equations, J. Comput. Phys., Volume 109 (1993) no. 2, pp. 222-237 | DOI | MR | Zbl

[12] F. Assous; P. Degond; J. Segré A particle-tracking method for 3D electromagnetic PIC codes on unstructured meshes, Comput. Phys. Commun., Volume 72 (1992), pp. 105-114 | DOI

[13] P. Degond; P.-A. Raviart On the paraxial approximation of the stationary Vlasov–Maxwell, Math. Models Methods Appl. Sci., Volume 3 (1993) no. 4, pp. 513-562 | DOI | MR | Zbl

[14] G. Laval; S. Mas-Gallic; P.-A. Raviart Paraxial approximation of ultrarelativistic intense beams, Numer. Math., Volume 69 (1994) no. 1, pp. 33-60 | DOI | MR | Zbl

[15] F. Filbet; E. Sonnendrücker Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Models Methods Appl. Sci., Volume 16 (2006) no. 5, pp. 763-791 | DOI | MR | Zbl

[16] M. A. Mostrom; D. Mitrovich; D. R. Welch The ARCTIC charged particle beam propagation code, J. Comput. Phys., Volume 128 (1996) no. 2, pp. 489-497 | DOI | Zbl

[17] P. A. Raviart; E. Sonnendrücker A hierarchy of approximate models for the Maxwell equations, Numer. Math., Volume 73 (1996) no. 3, pp. 329-372 | DOI | MR

[18] J. K. Boyd; E. P. Lee; S. S. Yu Aspects of three field approximations: Darwin, frozen, EMPULSE, 1985 (No. UCID-20453. Lawrence Livermore National Lab., CA, USA)

[19] S. Slinker; G. Joyce; J. Krall; R. F. Hubbard ELBA a three dimensional particle simulation code for high current beams, 1991 (Published in Proc. of the 14th Inter. Conf. Numer. Simul. Plasmas, Annapolis)

[20] A. Nouri Paraxial approximation of the Vlasov–Maxwell system: laminar beams, Math. Models Methods Appl. Sci., Volume 4 (1994) no. 02, pp. 203-221 | DOI | MR | Zbl

[21] F. Assous; J. Chaskalovic A new paraxial asymptotic model for the relativistic Vlasov–Maxwell equations, C. R. Mecan. Acad. Sci., Volume 340 (2012), pp. 706-714

[22] F. Assous; F. Tsipis Numerical paraxial approximation for highly relativistic beams, Comput. Phys. Commun., Volume 180 (2009) no. 7, pp. 1086-1097 | DOI | MR

[23] F. Assous; J. Chaskalovic Data mining techniques for scientific computing: application to asymptotic paraxial approximations to model ultra relativistic particles, J. Comput. Phys., Volume 230 (2011), pp. 4811-4827 | DOI

[24] G. Grimvall Characteristic quantities and dimensional analysis, Scientific Modeling and Simulations (S. Yip; T. D. de la Rubia, eds.) (Lecture Notes in Computational Science and Engineering), Volume 68, Springer, 2008, pp. 21-39 | DOI

[25] G. I. Barenblatt Dimensional Analysis, Gordon and Breach, New York, 1987

[26] F. Assous; J. Chaskalovic A paraxial asymptotic model for the coupled Vlasov–Maxwell problem in electromagnetics, J. Comput. Appl. Math., Volume 270 (2014), pp. 369-385 | DOI | MR | Zbl

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