Algorithms for numerical simulation of radio-frequency glow discharges

A fast and numerically stable technique has been developed for the numerical simulation of parallel-plate rf glow discharges, operating at frequencies exceeding ∼10 MHz. In such discharges the ion concentrations do not vary on the time scale of the rf cycle. Thus the transport equations for charged particles can be decoupled into a set of time-averaged and time-dependent equations. Rather than follow the evolution of the discharge following initiation, a solution of the time-averaged equations provides the steady-state solution directly. Once the time-averaged solution is known, a solution of the time-dependent equations yields the modulation of the time-dependent variables about their time-averaged value. Decoupling of the equations also reduces the number of variables at each step, thereby enabling optimization of matrix solutions for the system of nonlinear equations. Finally, good initial guesses are provided by solutions of simplified models. Electron kinetics and transport data are obtained by solving the zero-dimensional Boltzmann equation for electrons in conjunction with a model for the kinetics of excited states. Simulations have been performed for discharges in Ar and ${\mathrm{SF}}_6$. A typical case for an argon discharge is executed in 6 min on a Micro Vax II Computer as compared to twice as many minutes on a Cray computer for similar models reported by other workers. Results of the simulations are in agreement with those reported elsewhere and with measurements made using a tuned Langmuir probe.