Startseite Multi-Scale Paraxial Models to Approximate Vlasov–Maxwell Equations
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Multi-Scale Paraxial Models to Approximate Vlasov–Maxwell Equations

  • Franck Assous EMAIL logo und Yevgeni Furman
Veröffentlicht/Copyright: 3. Februar 2022

Abstract

Even today, solving numerically the time-dependent Vlasov–Maxwell equations is a challenging issue, and developing simpler but accurate approximate models is still worthwhile. Here, we propose a new family of paraxial asymptotic models that approximates the Vlasov–Maxwell system of equations. We introduce parameters in our models that allow us to handle relativistic cases, much slower beams or even non-relativistic cases. These models are derived by introducing a small parameter and provide static or quasi-static approximate equations that are 𝑛-th order accurate; 𝑛 may be chosen as required. Practically, one can select a model by determining the regime one is interested in and choosing the degree of accuracy needed.

MSC 2010: 41A60; 78A25; 78A30; 78A35; 34E05; 35A15

References

[1] F. Assous and J. Chaskalovic, Data mining techniques for scientific computing: Application to asymptotic paraxial approximations to model ultrarelativistic particles, J. Comput. Phys. 230 (2011), no. 12, 4811–4827. 10.1016/j.jcp.2011.03.005Suche in Google Scholar

[2] F. Assous and J. Chaskalovic, A paraxial asymptotic model for the coupled Vlasov-Maxwell problem in electromagnetics, J. Comput. Appl. Math. 270 (2014), 369–385. 10.1016/j.cam.2013.12.037Suche in Google Scholar

[3] F. Assous, P. Ciarlet and S. Labrunie, Mathematical Foundations of Computational Electromagnetism, Appl. Math. Sci. 198, Springer, Cham, 2018. 10.1007/978-3-319-70842-3Suche in Google Scholar

[4] F. Assous, P. Degond, E. Heintze, P.-A. Raviart and J. Segre, On a finite-element method for solving the three-dimensional Maxwell equations, J. Comput. Phys. 109 (1993), no. 2, 222–237. 10.1006/jcph.1993.1214Suche in Google Scholar

[5] F. Assous, P. Degond and J. Segré, A particle-tracking method for 3D electromagnetic PIC codes on unstructured meshes, Comput. Phys. Commun. 72 (1992), no. 2–3, 105–114. 10.1016/0010-4655(92)90142-LSuche in Google Scholar

[6] F. Assous and Y. Furman, A hierarchy of reduced models to approximate Vlasov–Maxwell equations for slow time variations, C. R. Mecanique 348 (2020), no. 12, 969–981. 10.5802/crmeca.50Suche in Google Scholar

[7] F. Assous and F. Tsipis, A PIC method for solving a paraxial model of highly relativistic beams, J. Comput. Appl. Math. 227 (2009), no. 1, 136–146. 10.1016/j.cam.2008.07.022Suche in Google Scholar

[8] F. Assous and F. Tsipis, Numerical paraxial approximation for highly relativistic beams, Comput. Phys. Commun. 180 (2009), no. 7, 1086–1097. 10.1016/j.cpc.2008.12.037Suche in Google Scholar

[9] G. I. Barenblatt, Dimensional Analysis, Gordon and Breach, New York, 1987. Suche in Google Scholar

[10] C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, CRC Press, Boca Raton, 2004. Suche in Google Scholar

[11] J. K. Boyd, E. P. Lee and S. Yu, Aspects of Three Field Approximations: Darwin, Frozen, EMPULSE (no. UCID-20453), Lawrence Livermore National Laboratory, Livermore, 1985. 10.2172/5673870Suche in Google Scholar

[12] B. Danly, G. Bekefi, R. Davidson, R. Temkin, T. Tran and J. Wurtele, Principles of gyrotron powered electromagnetic wigglers for free-electron lasers, IEEE J. Quantum Electron 23 (1987), 103–116. 10.1109/JQE.1987.1073205Suche in Google Scholar

[13] P. Degond, F. Hermeline, P. A. Raviart and J. Segré, Numerical modeling of axisymmetric electron beam devices using a coupled particle-finite element method, IEEE Trans. Magn. 27 (1991), no. 5, 4177–4180. 10.1109/20.105022Suche in Google Scholar

[14] P. Degond and P.-A. Raviart, An analysis of the Darwin model of approximation to Maxwell’s equations, Forum Math. 4 (1992), no. 1, 13–44. 10.1515/form.1992.4.13Suche in Google Scholar

[15] P. Degond and P.-A. Raviart, On the paraxial approximation of the stationary Vlasov–Maxwell system, Math. Models Methods Appl. Sci. 3 (1993), no. 4, 513–562. 10.1142/S0218202593000278Suche in Google Scholar

[16] F. Filbet and E. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Models Methods Appl. Sci. 16 (2006), no. 5, 763–791. 10.1142/S0218202506001340Suche in Google Scholar

[17] I. S. Grant and W. R. Phillips, Electromagnetism, John Wiley & Sons, New York, 2013. Suche in Google Scholar

[18] G. Grimvall, Characteristic quantities and dimensional analysis, Scientific Modeling and Simulations, Lect. Notes Comput. Sci. Eng. 68, Springer, Dordrecht (2008), 21–39. 10.1007/978-1-4020-9741-6_4Suche in Google Scholar

[19] W. J. Harris, U.S. Patent No. 3, 271,556. Washington, DC: U.S. Patent and Trademark Office, 1966. Suche in Google Scholar

[20] R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, CRC Press, Boca Raton, 1988. 10.1887/0852743920Suche in Google Scholar

[21] G. Laval, S. Mas-Gallic and P.-A. Raviart, Paraxial approximation of ultrarelativistic intense beams, Numer. Math. 69 (1994), no. 1, 33–60. 10.1007/s002110050079Suche in Google Scholar

[22] J. D. Lawson, The Physics of Charged-Particle Beams, Clarendon Press, Oxford, 1977. Suche in Google Scholar

[23] M. J. Madou, Manufacturing Techniques for Microfabrication and Nanotechnology (Vol. 2), CRC Press, Boca Raton, 2011. 10.1201/9781439895306Suche in Google Scholar

[24] R. B. Miller, An Introduction to the Physics of Intense Charged Particle Beams, Springer, Boston, 1984. 10.13182/FST84-A23161Suche in Google Scholar

[25] M. A. Mostrom, D. Mitrovich and D. R. Welch, The ARCTIC charged particle beam propagation code, J. Comput. Phys. 128 (1996), no. 2, 489–497. 10.1006/jcph.1996.0226Suche in Google Scholar

[26] A. Nouri, Paraxial approximation of the Vlasov–Maxwell system: Laminar beams, Math. Models Methods Appl. Sci. 4 (1994), no. 2, 203–221. 10.1142/S0218202594000121Suche in Google Scholar

[27] P.-A. Raviart, An analysis of particle methods, Numerical Methods in Fluid Dynamics (Como 1983), Lecture Notes in Math. 1127, Springer, Berlin (1985), 243–324. 10.1007/BFb0074532Suche in Google Scholar

[28] P.-A. Raviart and E. Sonnendrücker, A hierarchy of approximate models for the Maxwell equations, Numer. Math. 73 (1996), no. 3, 329–372. 10.1007/s002110050196Suche in Google Scholar

[29] M. Reiser, Theory and Design of Charged Particle Beams, John Wiley & Sons, New York, 2008. 10.1002/9783527622047Suche in Google Scholar

[30] S. Slinker, G. Joyce, J. Krall and R. F. Hubbard, ELBAr A three dimensional particle simulation code for high current beams, Proc. of the 14th Inter. Conf. Numer. Simul. Plasmas, Annapolis, 1991. Suche in Google Scholar

[31] T. M. Tran and J. S. Wurtele, Free-electron laser simulation techniques, Phys. Rep. 195 (1990), 1–21. 10.1016/0370-1573(90)90074-CSuche in Google Scholar

[32] A. Vlasov, On the kinetic theory of an assembly of particles with collective interaction, Russ. Phys. J. 9 (1945), 25–40. Suche in Google Scholar

Received: 2021-04-19
Revised: 2021-11-19
Accepted: 2022-01-03
Published Online: 2022-02-03
Published in Print: 2022-04-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 27.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/cmam-2021-0082/html
Button zum nach oben scrollen