Eulerian codes for the numerical solution of the Vlasov equation. (English) Zbl 1130.35370
Summary: A historical overview of Eulerian codes for the numerical solution of the Vlasov equation is presented, with special attention to characteristic methods. An evaluation of the performance of the cubic spline used for interpolation in the characteristic methods, with respect to other methods of interpolation, will be presented by comparing the solutions obtained by solving numerically different Vlasov-Poisson and Vlasov-Maxwell systems on a fixed Eulerian grid. Some recent developments of characteristic methods in two dimensions will be presented.
MSC:
| 35Q72 | Other PDE from mechanics (MSC2000) |
| 65M25 | Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs |
References:
| [1] | Knorr, G., Z Naturforsch, 18a, 1304 (1963) |
| [2] | Armstrong, T. P.; Harding, R. C.; Knorr, G.; Montgomery, D., Plasma physics, Methods in Computational Physics, vol. 9 (1969), Academic Press: Academic Press NY, p. 30 |
| [3] | Joyce, G.; Knorr, G.; Meier, H., J Comp Phys, 8, 53 (1971) · Zbl 0241.65081 |
| [4] | Knorr, G., J Comp Phys, 10, 165 (1973) |
| [5] | Shoucri, M.; Gagné, R., J Comp Phys, 25, 94 (1977) · Zbl 0377.65062 |
| [6] | Schumer, J. W.; Holloway, J. P., J Comp Phys, 144, 626 (1998) · Zbl 0936.76061 |
| [7] | Cheng, C. Z.; Knorr, G., J Comp Phys, 22, 330 (1976) |
| [8] | Shoucri, M.; Gagné, R., IEEE Plasma Sci, PS-6, 245 (1978) · Zbl 0378.76083 |
| [9] | Shoucri, M., J Phys, 40, 38 (1979) |
| [10] | Simon, A., Phys Fluids, 31, 217 (1988) |
| [11] | Ghizzo, A.; Izrar, B.; Bertrand, P.; Fijalkow, E.; Feix, M.; Shoucri, M., Phys Fluids, 31, 72 (1988) |
| [12] | Bertrand P, Ghizzo A, Feix M, Fijalkow E, Mineau P, Suh ND, et al. In: Doveil F, editor. Nonlinear Phenomena in Vlasov Plasmas, Cargèse Workshop; July 1988. p. 109.; Bertrand P, Ghizzo A, Feix M, Fijalkow E, Mineau P, Suh ND, et al. In: Doveil F, editor. Nonlinear Phenomena in Vlasov Plasmas, Cargèse Workshop; July 1988. p. 109. |
| [13] | Cheng, C. Z., J Comp Phys, 24, 348 (1977) |
| [14] | Shoucri M. In: Proceedings 10th annual Pittsburgh conference, modeling and simulation. Ed. Instrument Soc. of America, vol. 10; 1979. p. 1187.; Shoucri M. In: Proceedings 10th annual Pittsburgh conference, modeling and simulation. Ed. Instrument Soc. of America, vol. 10; 1979. p. 1187. |
| [15] | Shoucri, M., IEEE Plasma Sci, PS-7, 69 (1979) |
| [16] | Johnson, L. E., J Plasma Phys, 23, 433 (1980) |
| [17] | Rickman, J. D., IEEE Plasma Science, PS-10, 45 (1982) |
| [18] | Ghizzo, A.; Bertrand, P.; Shoucri, M.; Johnston, T.; Feix, M., J Comp Phys, 90, 431 (1990) · Zbl 0702.76080 |
| [19] | Bertrand, P.; Ghizzo, A.; Johnston, T.; Shoucri, M.; Fijalkow, E., Phys Fluids, B2, 1028 (1990) |
| [20] | Johnston, T.; Bertrand, P.; Ghizzo, A.; Shoucri, M.; Fijalkow, E., Phys Fluids, B4, 2523 (1992) |
| [21] | Ghizzo, A.; Shoucri, M.; Bertrand, P.; Johnston, T.; Lebas, J., J Comp Phys, 108, 373 (1993) · Zbl 0791.76056 |
| [22] | Shoucri, M.; Bertrand, P.; Ghizzo, A.; Lebas, J.; Johnston, T., Phys Letters A, 156, 76 (1991) |
| [23] | Ghizzo, A.; Bertrand, P.; Lebas, J.; Shoucri, M.; Johnston, T., J Comp Phys, 102, 417 (1992) |
| [24] | Bertrand, P.; Ghizzo, A.; Karttunen, S.; Pättikangas, T.; Salomaa, R.; Shoucri, M., Phys Rev E, 49, 5656 (1994) |
| [25] | Ghizzo, A.; Bertrand, P.; Bégué, M. L.; Johnston, T., IEEE Plasma Sci, 24, 370 (1996) |
| [26] | Bégué, M. L.; Ghizzo, A.; Bertrand, P., J Comp Phys, 151, 458 (1999) · Zbl 0956.76068 |
| [27] | Ghizzo, A.; Bertrand, P.; Shoucri, M.; Johnston, T.; Fijalkow, E., Nucl Fusion, 32, 45 (1992) |
| [28] | Joyce G, Krall J, Slinker S. NRL Memorandum report 6772; June 1991.; Joyce G, Krall J, Slinker S. NRL Memorandum report 6772; June 1991. |
| [29] | Ruhl, H.; Mulser, P., Phys Lett A, 205, 388 (1995) |
| [30] | Brunner, S.; Valeo, E., Phys Rev Lett, 93, 145003 (2004) |
| [31] | Strozzi, D.; Williams, E.; Langdon, A. B.; Bers, A., Phys Plasmas, 14, 013104 (2007) |
| [32] | Manfredi, G.; Shoucri, M.; Feix, M.; Bertrand, P., J Comp Phys, 121, 298 (1995) · Zbl 0840.76098 |
| [33] | Manfredi, G.; Shoucri, M.; Shkarofsky, I.; Ghizzo, A., Fusion Technol, 29, 24 (1996) |
| [34] | Manfredi, G.; Shoucri, M.; Dendy, R. O.; Ghizzo, A., Phys Plasmas, 3, 202 (1996) |
| [35] | Manfredi, G.; Shoucri, M.; Bertrand, P.; Ghizzo, A.; Knorr, G., Phys Scripta, 58, 159 (1998) |
| [36] | Shoucri, M.; Lebas, J.; Knorr, G.; Bertrand, P.; Ghizzo, A., Phys Scripta, 55, 617 (1997) |
| [37] | Shoucri, M.; Manfredi, G.; Bertrand, P.; Ghizzo, A.; Knorr, G., J Plasma Phys, 61, 191 (1999) |
| [38] | Karttunen, S.; Pättikangas, T.; Salomaa, R.; Shoucri, M., Phys Scripta, 60, 356 (1999) |
| [39] | Grangirard, V.; Brunetti, M., J Comp Phys, 217, 395 (2006) |
| [40] | Candy, J.; Waltz, R. E., Phys Rev Lett, 91, 045001 (2003) |
| [41] | Watanabe, T.-H.; Sugama, H.; Sato, T., J Phys Soc Jpn, 70, 3565 (2001) |
| [42] | Klimas, A. J.; Farrell, W. M., J Comp Phys, 110, 150 (1994) · Zbl 0790.76064 |
| [43] | Watanabe, T.-H.; Sugama, H., Trans Theory Stat, 34, 287 (2005) · Zbl 1101.82037 |
| [44] | Nakamura, T.; Yabe, T., Comp Phys Commun, 120, 122 (1999) · Zbl 1001.82003 |
| [45] | Fijalkow, E., Comp Phys Commun, 116, 319 (1999) · Zbl 1019.76035 |
| [46] | Horne, R. B.; Freeman, M. P., J Comp Phys, 171, 182 (2001) · Zbl 0983.78021 |
| [47] | Mangeney, A.; Califano, F.; Cavazzoni, C.; Travnicek, P., J Comp Phys, 179, 475 (2002) |
| [48] | Arber, T. D.; Vann, R. G.L., J Comput Phys, 180, 339 (2002) · Zbl 1001.82105 |
| [49] | Pohn E, Shoucri M. Commun Nonlinear Sci Numer Simul, this issue.; Pohn E, Shoucri M. Commun Nonlinear Sci Numer Simul, this issue. |
| [50] | Demeio, L.; Zweifel, P. F., Phys Fluids, B3, 501 (1991) |
| [51] | Califano, F.; Pegoraro, F., Trans Theory Stat Phys, 32, 413 (2003) · Zbl 1029.82042 |
| [52] | Krane, B.; Christopher, I.; Shoucri, M.; Knorr, B., Phys Rev Lett, 80, 4422 (1998) |
| [53] | Sonnendrücker, E.; Barnard, J., Nucl Instrum Meth Phys Res A, 464, 653 (2001) |
| [54] | Fibet, F.; Sonnendrücker, E., Comp Phys Commun, 150, 24 (2003) |
| [55] | Todo, Y.; Sato, T.; Watanabe, K.; Watanabe, T. H.; Horiuchi, R., Phys Plasmas, 2, 2711 (1995) |
| [56] | Makar, P.; Karpik, S. R., Mon Weath Rev, 124, 182 (1995) |
| [57] | Sonnendrücker, E.; Roche, J.; Bertrand, P.; Ghizzo, A., J Comp Phys, 149, 201 (1998) |
| [58] | Reprinted from Comp. Phys. Comm., Vol. 154, M. Shoucri, H. Gerhauser, K.-H. Finken, Integration of the Vlasov equation along characteristics in one and two dimensions, p. 68-69, Copyright 2003, with permission from Elsevier.; Reprinted from Comp. Phys. Comm., Vol. 154, M. Shoucri, H. Gerhauser, K.-H. Finken, Integration of the Vlasov equation along characteristics in one and two dimensions, p. 68-69, Copyright 2003, with permission from Elsevier. · Zbl 1196.76112 |
| [59] | Batishchev, O. V.; Shoucri, M.; Batishcheva, A.; Shkarofsky, I., J Plasma Phys, 61, 347 (1999) |
| [60] | Bertrand P, Shoucri M, Ghizzo A, Izrar B, Feix M, et al. CCFM Report No. 220; January 1987.; Bertrand P, Shoucri M, Ghizzo A, Izrar B, Feix M, et al. CCFM Report No. 220; January 1987. |
| [61] | Pohn, E.; Shoucri, M.; Kamelander, G., Comp Phys Commun, 137, 396 (2001) · Zbl 1097.76612 |
| [62] | Shoucri, M.; Pohn, E.; Knorr, G.; Bertrand, P.; Kamelander, G., Phys Plasmas, 7, 2517 (2000) |
| [63] | Shoucri, M.; Gerhauser, H.; Finken, K.-H., Comp Phys Commun, 164, 138 (2004) · Zbl 1196.82031 |
| [64] | Huot, F.; Ghizzo, A.; Bertrand, P.; Sonnendrücker, E., J Comp Phys, 185, 512 (2003) · Zbl 1073.82039 |
| [65] | Shoucri, M., The method of characteristics for the numerical solution of hyperbolic differential equations, (Bianco, S. J., Computer Physics Research Trends (2007), Nova Science Publishers Inc: Nova Science Publishers Inc NY) · Zbl 1196.82032 |
| [66] | Shoucri, M.; Matte, J. P.; Côté, A., J Phys D: Appl Phys, 36, 2083 (2003) |
| [67] | Nonlinear Vlasov plasmas. In: F. Doveil, editor. Proceedings of the Cargèse International Workshop; July 1988.; Nonlinear Vlasov plasmas. In: F. Doveil, editor. Proceedings of the Cargèse International Workshop; July 1988. |
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