The finite volume method solution of the radiative transfer equation for photon transport in non-thermal gas discharges: application to the calculation of photoionization in streamer discharges

This paper presents the development of a direct accurate numerical method to solve the monochromatic radiative transfer equation (RTE) based on a finite volume method (FVM) and its application to the simulation of streamer propagation. The validity of the developed model is demonstrated by performing direct comparisons with results obtained using the classic integral model. Comparisons with approximate solutions of the RTE (Eddington and SP3 models) are also carried out. Specific validation comparisons are presented for an artificial source of radiation with a Gaussian shape. The reported results demonstrate that whatever the value of the absorption coefficient, the results obtained using the direct FVM are in excellent agreement with the reference integral model with a significantly reduced computation time. When the absorption coefficient is high enough, the Eddington and SP3 methods are as accurate and become faster than the FVM. However, when the absorption coefficient decreases, approximate methods become less accurate and more computationally expensive than the FVM. Then the direct finite volume and the SP3 models have been applied to the calculation of photoionization in a double-headed streamer at ground pressure. For high values of the absorption coefficient, positive and negative streamers calculated using the SP3 model and the FVM for the photoionization source term are in excellent agreement. As the value of the absorption coefficient decreases, discrepancies between the results obtained with the finite volume and the SP3 models increase, and these differences increase as the streamers advance. For low values of the absorption coefficient, the use of the SP3 model overestimates the electron density and underestimates the photoionization source term in both streamers in comparison with the FVM. As a consequence, for low values of the absorption coefficient, positive and negative streamers calculated using the SP3 model for the photoionization source term propagate more slowly than those calculated using the FVM.