Geodesic
An algorithm for the three-dimensional modeling of ultrarelativistic beams
Numerical methods and programming, Tome 8 (2007) no. 4, pp. 352-359.

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An algorithm for modeling the interaction charged particle beam effects in the three-dimensional ultrarelativistic case is considered. An algorithm to compute initial and boundary conditions when introducing an artificial potential is proposed. Some numerical test results for prescribing boundary conditions and for solving the problem of beam focusing at the region center are discussed.
Keywords: ultrarelativistic beams, electron-positron colliding beams, numerical methods, mathematical simulation, supercollider. 
@article{VMP_2007_8_4_a4,
     author = {M. A. Boronina and V. A. Vshivkov and E. B. Levichev and S. A. Nikitin and V. N. Snytnikov},
     title = {An algorithm for the three-dimensional modeling of ultrarelativistic beams},
     journal = {Numerical methods and programming},
     pages = {352--359},
     publisher = {mathdoc},
     volume = {8},
     number = {4},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2007_8_4_a4/}
}
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M. A. Boronina; V. A. Vshivkov; E. B. Levichev; S. A. Nikitin; V. N. Snytnikov. An algorithm for the three-dimensional modeling of ultrarelativistic beams. Numerical methods and programming, Tome 8 (2007) no. 4, pp. 352-359. http://geodesic.mathdoc.fr/item/VMP_2007_8_4_a4/