Numerical Fokker-Planck calculations in nonuniform grids

Physics of Plasmas

Volumn 8, Issue 5 II, 2001, Pages 1903-1909

  Bizarro, João P S ^a   Rodrigues, Paulo ^a  

^a INSTITUTO SUPERIOR TÉCNICO   (Portugal)

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EID: 0035333159     PISSN: 1070664X     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1348331     Document Type: Article

References (27)

1. S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
2. N. O. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1984).
3. M. N. Rosenbluth, W. M. MacDonald, and D. L. Judd, Phys. Rev. 107, 1 (1957).
4. W. M. MacDonald, M. N. Rosenbluth, and W. Chuck, Phys. Rev. 107, 350 (1957).
5. T. H. Kho, Phys. Rev. A 32, 666 (1985).
6. J. S. Chang and G. Cooper, J. Comput. Phys. 6, 1 (1970).
7. E. M. Epperlein, J. Comput. Phys. 112, 291 (1994).
8. N. J. Fisch, Rev. Mod. Phys. 59, 175 (1987).
9. C. F. F. Karney and N. J. Fisch, Nucl. Fusion 12, 1549 (1981).
10. C F. F. Karney, Comput. Phys. Rep. 4, 183 (1986).
11. M. G. McCoy, A. A. Mirin, and J. Killeen, Comput. Phys. Commun. 24, 37 (1981).
12. M. R. O'Brien, M. Cox, and D. F. H Start, Comput. Phys. Commun. 40, 123 (1986).
13. M. Shoucri and I Shkarofsky, Comput. Phys. Commun. 82, 287 (1994).
14. Y. Peysson and M. Shoucri, Comput. Phys. Commun. 109, 55 (1998).
15. J. P. S. Bizarro and P. Rodrigues, Nucl. Fusion 37, 1509 (1997).
16. E. W. Larsen, C. D. Levermore, G. C. Pomraning, and J. G. Sanderson, J. Comput. Phys. 61, 359 (1985).
17. A. N. Drozdov and M. Morillo, Phys. Rev E 54, 931 (1996).
18. Basically, this is because in 2-D, and contrary to what happens in 1-D, the vanishing of the generalized flux at the boundaries does not imply, in quasiequilibrium conditions, that it must be zero everywhere.
19. D. L. Scharfetter and H. K. Gummel, IEEE Trans. Electron Devices ED-16, 64 (1969).
20. J.P. Boeuf, Phys. Rev. A 36, 2782 (1987).
21. J. D. P. Passchier and W. J. Goedheer, J. Appl. Phys. 74, 3744 (1993).
22. Without going into details, it can be noted that the weighting coefficients introduced in Ref. 6 (using uniform grids, and implicitly taking the friction and diffusion FP coefficients constant within grid cells) remain unchanged if, instead of computing them for a zero (quasiequilibrium) generalized flux, they are retrieved assuming the latter is constant within each cell. The constancy of the flux between mesh points (as well as of the convection and diffusion coefficients) is precisely the approximation made in Ref. 19. The fact that the zero-flux condition can actually be relaxed probably explains why, as in Refs. 9 and 15, the generalization of such weighting coefficients to 2-D (where that condition is not a necessary one for equilibrium) has been successful.
23. W. R. Briley and H. McDonald, J. Comput. Phys. 34, 54 (1980).
24. For Coulombian collisional tranport, it can be shown that Eq. (10) implies energy is strictly conserved, in accordance with Ref. 7. Note, furthermore, that Eqs. (9) and (10) are a generalization of the results on particle and energy conservation given in that work, which have been derived specifically for the FP equation describing Coulomb collisions.
25. In some cases, such as in Ref. 7, analytically establishing energy conservation may need successive integrations by parts (whose discrete counterparts may not necessarily be so-called summations by parts, as used in Ref. 10), and this may contribute additional terms to the total error in the energy integral of Eq. (4).
26. Note that Eq. (22) differs from the expression given in Ref. 15 for the weighting coefficients (apart from different signs m the definition of the flux) solely in the way the derivative in Eq. (19) is calculated, the expression adopted here demanding no knowledge of the friction and diffusion functions in the centers of grid cells (using instead their values at the grid boundaries, where they are needed for the flux), making it more computationally friendly.
27. In this particular case, the domain where the FP equation is to be solved ranges from minus to plus infinity (and not from zero to infinity, as assumed in Sec. II), but the necessary modifications are evident. Moreover, in this as in all examples discussed Sec. III, all variables are duly normalized and specific details of the respective physical models are easily retrieved from the references cited.