Numerical study by a controllability method for the calculation of the time-periodic solutions of the Maxwell and Vlasov-Maxwell systems
Bostan, Mihai
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001), p. 165-189 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

The topic of this paper is the numerical analysis of time periodic solution for electro-magnetic phenomena. The Limit Absorption Method (LAM) which forms the basis of our study is presented. Theoretical results have been proved in the linear finite dimensional case. This method is applied to scattering problems and transport of charged particles.

Publié le : 2001-01-01
Classification:  34A30,  35G60,  35F30 
@article{M2AN_2001__35_1_165_0,
     author = {Bostan, Mihai},
     title = {Numerical study by a controllability method for the calculation of the time-periodic solutions of the Maxwell and Vlasov-Maxwell systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {35},
     year = {2001},
     pages = {165-189},
     mrnumber = {1811986},
     zbl = {1041.78005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2001__35_1_165_0}
}

Bostan, Mihai. Numerical study by a controllability method for the calculation of the time-periodic solutions of the Maxwell and Vlasov-Maxwell systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) pp. 165-189. http://gdmltest.u-ga.fr/item/M2AN_2001__35_1_165_0/

[1] A. Arsenev, Global existence of a weak solution of Vlasov's system of equations. USSR Comp. Math. Math. Phys. 15 (1975) 131-143.

[2] K. Asano and S. Ukai, On the Vlasov-Poisson limit of the Vlasov-Maxwell equation. Pattern and waves. Qualitative analysis of nonlinear differential equations. Stud. Math. Appl. 18 (1986) 369-383. | Zbl 0623.35059

[3] N. Ben.Abdallah, Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system,. Math. Meth. Appl. Sci. 17 (1994) 451-476. | Zbl 0806.35172

[4] M. Bezard, Boundary value problems for the Vlasov-Maxwell system, in Semin. Équ. Deriv. Partielles, Ec. Polytech., Cent. Math., Palaiseau Semi 1992-1993, Exp. No. 4 (1993) 17. | Numdam | Zbl 0878.35107

[5] B. Bodin, Modélisation et simulation numérique du régime de Child-Langmuir. Thèse de l'École Polytechnique, Palaiseau (1995).

[6] M. Bostan and F. Poupaud, Periodic solutions of the Vlasov-Poisson system with boundary conditions. C. R. Acad. Sci. Paris, Sér. I 325 (1997) 1333-1336. | Zbl 0895.35079

[7] M. Bostan and F. Poupaud, Periodic solutions of the Vlasov-Poisson system with boundary conditions. Math. Mod. Meth. Appl. Sci. 10 (1998) 651-672. | Zbl 1019.76047

[8] M. Bostan and F. Poupaud, Periodic solutions of the 1D Vlasov-Maxwell system with boundary conditions. Math. Meth. Appl. Sci. 23 (2000) 1195-1221. | Zbl 0965.35010

[9] M.O. Bristeau, R. Glowinski and J. Périaux, Controllability methods for the computation of time periodic solutions; application to scattering. J. Comp. Phys. 147 (1998) 265-292. | Zbl 0926.65054

[10] J.P. Cioni, Résolution numérique des équations de Maxwell instationnaires par une méthode de volumes finis. Ph.D., Université de Nice Sophia-Antipolis (1995).

[11] J.P. Cioni, L. Fezoui and D. Issautier, High-order upwind schemes for solving time-domain Maxwell equation. La Recherche Aérospatiale No. 5 (1994) 319-328.

[12] P. Degond, Regularité de la solution des équations cinétiques en physiques de plasmas, in Semin. Équ. Dériv. Partielles 1985-1986, Exp. No. 18 (1986) 11. | Numdam | Zbl 0613.35070

[13] P. Degond, Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity. Math. Methods Appl. Sci. 8 (1986) 533-558. | Zbl 0619.35088

[14] P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions. Ann. Sci. Ec. Norm. Super. IV. Ser. 19 (1986) 519-542. | Numdam | Zbl 0619.35087

[15] R.J. Diperna and P.L. Lions, Global weak solutions of Vlasov-Maxwell system. Comm. Pure Appl. Math. XVII (1989) 729-757. | Zbl 0698.35128

[16] C. Greengard and P.A. Raviart, A boundary value problem for the stationary Vlasov-Poisson system. Comm. Pure Appl. Math. XLIII (1990) 473-507. | Zbl 0721.35084

[17] Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions. Comm. Math. Phys. 154 (1993) 245-263. | Zbl 0787.35072

[18] Y. Guo, Regularity for the Vlasov equation in a half space. Indiana Univ. Math. J. 43 (1994) 255-320. | Zbl 0799.35031

[19] P.L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Invent. Math. 105 (1991) 415-430. | Zbl 0741.35061

[20] R. Löhner and J. Ambrosiano, A finite element solver for the Maxwell equations, in GAMNI-SMAI conference on numerical methods for the solution of Maxwell equations, Paris (1989).

[21] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in 3 dimensions for general initial data. J. Diff. Eq. 95 (1992) 281-303. | Zbl 0810.35089

[22] F. Poupaud, Boundary value problems for the stationary Vlasov-Maxwell system. Forum Math. 4 (1992) 499-527. | Zbl 0785.35020