February  2015, 8(1): 185-221. doi: 10.3934/dcdss.2015.8.185

Application of Lie transform techniques for simulation of a charged particle beam

1. 

Inria Nancy-Grand Est, CALVI Project, & IRMA (UMR CNRS 7501), Université de Strasbourg, 7, rue René-Descartes, 67084, Strasbourg, France

Received  February 2013 Revised  May 2013 Published  July 2014

We study a Lie Transform method for a charged beam under the action of a radial external electric field. The aim of the Lie transform method that is used here is to construct a change of variable which transforms the 2D kinetic problem into a 1D problem. This reduces the dimensionality of the problem and make it easier to solve numerically. After applying the Lie transform method, we truncate the expression of the characteristics of the Vlasov equation and the expression of the Poisson equation in the Lie coordinate system and we develop a numerical method for solving the truncated model and we study its efficiency for the simulation of long time beam evolution.
Citation: Mathieu Lutz. Application of Lie transform techniques for simulation of a charged particle beam. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 185-221. doi: 10.3934/dcdss.2015.8.185
References:
[1]

J. A. Brizard, Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks, Physics of Plasmas, 2 (1995), 459-471. doi: 10.1063/1.871465.

[2]

N. Crouseilles, M. Lemou and F. Méhats, Asymptotic preserving schemes for highly oscillatory kinetic equations, J. Comp. Phys., 248 (2013), 287-308. doi: 10.1016/j.jcp.2013.04.022.

[3]

D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations, Physics of Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113.

[4]

F. Filbet and E. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Mathematical Models and Methods in Applied Sciences, 16 (2006), 763-791. doi: 10.1142/S0218202506001340.

[5]

E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Mathematical Models and Methods in Applied Sciences, 19 (2009), 175-197. doi: 10.1142/S0218202509003395.

[6]

E. Frénod and E. Sonnendrücker, The finite larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243.

[7]

E. A. Frieman and L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria, Physics of Fluids, 25 (1982), 502-508. doi: 10.1063/1.863762.

[8]

T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence, Physics of Fluids, 31 (1988), 2670-2673. doi: 10.1063/1.866544.

[9]

T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas, Physics of Fluids, 31 (1988), 1940-1948. doi: 10.1063/1.866641.

[10]

R. G. Littlejohn, A guiding center Hamiltonian: A new approach, Journal of Mathematical Physics, 20 (1979), 2445-2458. doi: 10.1063/1.524053.

[11]

R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Physics of Fluids, 24 (1981), 1730-1749. doi: 10.1063/1.863594.

[12]

R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates, Journal of Mathematical Physics, 23 (1982), 742-747. doi: 10.1063/1.525429.

[13]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer, 1999. doi: 10.1007/978-0-387-21792-5.

[14]

K. R. Meyer and G. R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Sciences, 90, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4757-4073-8.

[15]

P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.

[16]

F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales, Plasma Physics and Controlled Fusion, 50 (2008), 065014. doi: 10.1088/0741-3335/50/6/065014.

[17]

F. I. Parra and P. J. Catto, Gyrokinetic equivalence, Plasma Physics and Controlled Fusion, 51 (2009), 065002. doi: 10.1088/0741-3335/51/6/065002.

[18]

F. I. Parra and P. J. Catto, Turbulent transport of toroidal angular momentum in low flow gyrokinetics, Plasma Physics and Controlled Fusion, 52 (2010), 045004. doi: 10.1088/0741-3335/52/4/045004.

[19]

H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, General gyrokinetic equations for edge plasmas, Contributions to Plasma Physics, 46 (2006), 477-489. doi: 10.1002/ctpp.200610034.

show all references

References:
[1]

J. A. Brizard, Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks, Physics of Plasmas, 2 (1995), 459-471. doi: 10.1063/1.871465.

[2]

N. Crouseilles, M. Lemou and F. Méhats, Asymptotic preserving schemes for highly oscillatory kinetic equations, J. Comp. Phys., 248 (2013), 287-308. doi: 10.1016/j.jcp.2013.04.022.

[3]

D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations, Physics of Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113.

[4]

F. Filbet and E. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Mathematical Models and Methods in Applied Sciences, 16 (2006), 763-791. doi: 10.1142/S0218202506001340.

[5]

E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Mathematical Models and Methods in Applied Sciences, 19 (2009), 175-197. doi: 10.1142/S0218202509003395.

[6]

E. Frénod and E. Sonnendrücker, The finite larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243.

[7]

E. A. Frieman and L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria, Physics of Fluids, 25 (1982), 502-508. doi: 10.1063/1.863762.

[8]

T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence, Physics of Fluids, 31 (1988), 2670-2673. doi: 10.1063/1.866544.

[9]

T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas, Physics of Fluids, 31 (1988), 1940-1948. doi: 10.1063/1.866641.

[10]

R. G. Littlejohn, A guiding center Hamiltonian: A new approach, Journal of Mathematical Physics, 20 (1979), 2445-2458. doi: 10.1063/1.524053.

[11]

R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Physics of Fluids, 24 (1981), 1730-1749. doi: 10.1063/1.863594.

[12]

R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates, Journal of Mathematical Physics, 23 (1982), 742-747. doi: 10.1063/1.525429.

[13]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer, 1999. doi: 10.1007/978-0-387-21792-5.

[14]

K. R. Meyer and G. R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Sciences, 90, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4757-4073-8.

[15]

P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.

[16]

F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales, Plasma Physics and Controlled Fusion, 50 (2008), 065014. doi: 10.1088/0741-3335/50/6/065014.

[17]

F. I. Parra and P. J. Catto, Gyrokinetic equivalence, Plasma Physics and Controlled Fusion, 51 (2009), 065002. doi: 10.1088/0741-3335/51/6/065002.

[18]

F. I. Parra and P. J. Catto, Turbulent transport of toroidal angular momentum in low flow gyrokinetics, Plasma Physics and Controlled Fusion, 52 (2010), 045004. doi: 10.1088/0741-3335/52/4/045004.

[19]

H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, General gyrokinetic equations for edge plasmas, Contributions to Plasma Physics, 46 (2006), 477-489. doi: 10.1002/ctpp.200610034.

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