Topic/Type: 2.1 MHD, EHD & other fluid methods, Poster

### Fast time-relaxation algorithm to solve plasma fluid-equations

J.Greg?rio1, 2, C.Boisse-Laporte2, L.L.Alves1

1 Instituto de Plasmas e Fus?o Nuclear, Instituto Superior T?cnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
2 Laboratoire de Physique des Gaz et des Plasmas, Universit? de Paris-Sud, 91405 Orsay, France

Low-temperature, partially ionized plasmas are complex systems formed by several kinds of charged and neutral species, whose populations (usually out of equilibrium) are deeply interconnected via (i) collisional events and (ii) long-range interactions mediated by electromagnetic fields (responsible for plasma excitation and modified by its charged particles). In this context, the study of the plasma dynamics (including the production / destruction of species and their transport in the presence of fields) is a formidable problem, which can only be correctly described through a non-linear, self-consistent description.
In general, the self-consistent modeling of low-temperature plasmas implies the use of iterative, non-implicit algorithms, due to the non-linear features of the problem aggravated by its size, in terms of both the number of variables considered and the workspace dimension. Often, these are time-relaxation algorithms presenting the inconvenient of being very time consuming, as they advance the different equations to solve in a sequential manner, within very small time-steps to ensure stability.
In this paper, we present a fast time-relaxation, quasi-implicit algorithm to solve plasma fluid-type equations, built in such a way that the different equations describing the transport of species with the plasma are managed within almost independent calculation modules. The algorithm is applied to the modeling of atmospheric-pressure micro-plasmas in argon, produced by a continuous microwave excitation (2.45 GHz frequency). The plasmas are created within the 50-200~$\small \mu$m slit separating two metal blades (6-14 mm width), which correspond to the end-gap of a microstrip-like transmission line [1]. The model solves the one-dimensional (between the metal blades), stationary transport equations for electrons, positive ions Ar$\small ^+$ and Ar$\small ^+_2$, excited species Ar(4s) and the electron mean energy, together with Poisson?s equation for the space-charge electrostatic field, Maxwell?s equations for the microwave excitation field and the gas thermal energy equation [2].
The electron continuity and momentum-transfer equations are solved implicitly, until full convergence, given the initial profiles of the densities and the fields, and imposing the electron density at the boundary axis. Only then, the algorithm solves the transport equations for the ions, the excited species and the gas thermal energy, together with Poisson?s equation, replacing Neumann by equivalent-Dirichlet boundary conditions, to ensure stability. After convergence of these two modules, the algorithm solves implicitly the equations for the electron mean energy and the microwave field, and calculates the input power as the eigenvalue that satisfies energy boundary conditions. Equations are solved in a non-uniform mesh of 350 points, for a convergence criterion that imposes relative errors smaller than 10$\small ^{-7}$ for all calculated quantities, ensuring run times of 30-60 minutes on a 3.0 GHz CPU.

[1] J. Greg?rio, O. Leroy, P. Leprince, L.L. Alves, C. Boisse-Laporte, IEEE Trans. Plasma Sci. 37, 797 (2009).

[2] J. Greg?rio, C. Boisse-Laporte, L.L. Alves, submitted to Eur. Phys. J. Appl. Phys. (2009).