Topic/Type: 2.2 Kinetic methods, Particle-In-Cell and Vlasov, Poster

Efficient energy-conserving algorithm for electron-electron collisions in the P1 approximation to the Boltzmann equation

M. Capitelli 1, G. Colonna2, G. Coppa3, A. D'Angola 4, R. Faticato 4

1 Universit? di Bari, via Orabona, 4 - 70126 Bari, Italy
2 CNR-IMIP Bari, via Amendola 122/D - 70126 Bari, Italy
3 Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino - Italy
4 Universit? della Basilicata, via dell?Ateneo Lucano, 10 - 85100 Potenza, Italy

The knowledge of the electron energy distribution function (eedf) in weakly ionized gases is a fundamental aspect in modeling plasma chemical processes, ranging from gas discharges [1] to re-entry of space vehicles [2]. Inelastic and superelastic collisions of free electrons with neutrals result in large deviations of the eedf from a Maxwellian. In order to reduce the computational effort, the contribution of e-e collisions in electron kinetics is usually neglected or alternatively a Maxwellian distribution is adopted. However, the synergic interaction between electron-electron and superelastic collisions strongly affects the eedf, especially in presence of electronically metastable states [3]. In these conditions the eedf must be calculated by solving the Boltzmann equation. In particular, the P1 approximation [4], resulting in a Fokker-Planck equation, is commonly used in the case of weak electric field. In the equation, elastic, inelastic and superelastic collisions give a linear contribution, while e-e collisions introduce nonlinear terms. Self-consistent coupling of the kinetics of free electrons and heavy particles makes the global problem non-linear, because the population of atoms in excited states, entering in the e-neutral collisional terms, depends on the eedf, through e-neutrals chemical rates.
The commonly used algorithm to calculate the eedf has been developed by S.D. Rockwood in 1973 [4]. In fact, the method is rather time-consuming in evaluating the nonlinear contribution of e-e collisions. Moreover, conservation of the total electron energy in e-e collisions is formally imposed but not fulfilled when numerical time integration is carried out. In this paper we present an efficient numerical algorithm which takes into account e-e nonlinear terms by using sparse matrices, preserving exactly the total electron energy. These are critical points in self-consistent models, especially in fluid dynamic simulations, where the total energy equation is considered and the Boltzmann equation must be solved many times.
The performance of the Rockwood\'s and of the present algorithms have been compared, obtaining (for 400 energy groups) a gain in the computation time of a factor 6 when only e-e collisions are considered and a factor 25 when all the processes are considered. The factor increases with the number energy groups. Moreover, the total electron energy is conserved under the required tolerance.

[1] G. Capriati, J. P. B?uf, and M. Capitelli, Plasma Chemistry and Plasma Processing, 13(3):499?519, 1993.

[2] G. Colonna and M. Capitelli, Journal of Thermophysics and Heat Transfer, 22(3):414?423, 2008.

[3] G. Colonna and M. Capitelli, Journal of Physics D: Applied Physics, 34:1812?1818, 2001.

[4] S. D. Rockwood, Physical Review A, 8(5):2348?2360, 1973.