The use of an improved Eddington approximation to facilitate the calculation of photoionization in streamer discharges

In the simulation of streamer discharge propagation, classical integral methods used to calculate the photoionization source term are computationally very expensive. In this work, a new approach based on the direct solution of an approximate radiative transfer equation is developed. Different approximations of the radiative transfer equation are discussed and tested for typical conditions encountered in streamer discharges. An improved Eddington approximation is shown to be very accurate to calculate the photoionization term for a Gaussian emission source term with a half-width length of the order of 0.02 cm when the absorption coefficient of the gas is higher than or equal to 50 cm^-1. For steeper gradients of the source term, good agreement is obtained for higher values of the absorption coefficient. Furthermore, the computation time of the improved Eddington method is four orders of magnitude less than with the usual integral method. For streamer propagation in air at atmospheric pressure, the absorption coefficient is shown to be of the order of 130 cm^-1 which validates the use of the improved Eddington approximation to calculate the photoionization term. Finally, two-dimensional calculations of a positive streamer discharge in air at atmospheric pressure in plane-plane geometry with the improved Eddington approximation are presented.