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In this paper, we give a detailed review of recent work carried out on the numerical characterization of non-thermal gas discharge plasmas in air at atmospheric pressure. First, we briefly describe the theory of discharge development for dielectric barrier discharges, which is central to the production of non-equilibrium plasma, and we present a hydrodynamic model to approximate the evolution of charge densities. The model consists of the continuity equations for electrons, positive and negative ions coupled to Poisson's equation for the electric field. We then describe features of the finite element flux corrected transport algorithm, which has been developed to specifically aim for accuracy (no spurious diffusion or oscillations), efficiency (through the use of unstructured grids) and ease of extension to complex 3D geometries in the framework of the hydrodynamic model in gas discharges. We summarize the numerical work done by other authors who have applied different methods to various models and then we present highlights of our own work, which includes code validation, comparisons with existing results and modelling of radio frequency systems, dc discharges, secondary effects such as photoionization and plasma production in the presence of dielectrics. The extension of the code to 3D for more realistic simulations is demonstrated together with the adaptive meshing technique, which serves to achieve higher efficiency. Finally, we illustrate the versatility of our scheme by using it to simulate the transition from non-thermal to thermal discharges.We conclude that numerical modelling and, in particular, the extension to 3D can be used to shed new light on the processes involved with the production and control of atmospheric plasma, which plays an important role in a host of emerging technologies.

An improved finite-element flux-corrected transport (FE-FCT) scheme, which was demonstrated in one dimension by the authors, is now extended to two dimensions and applied to gas discharge problems. The low-order positive ripple-free scheme, required to produce a FCT algorithm, is obtained by introducing diffusion to the high-order scheme (two-step Taylor-Galerkin). A self-adjusting variable diffusion coefficient is introduced, which reduces the high-order scheme to the equivalent of the upwind difference scheme, but without the complexities of an upwind scheme in a finite-element setting. Results are presented which show that the high-order scheme reduces to the equivalent of upwinding when the new diffusion coefficient is used. The proposed FCT scheme is shown to give similar results in comparison to a finite-difference time-split FCT code developed by Boris and Book. Finally, the new method is applied for the first time to a streamer propagation problem in its two-dimensional form.

A new approach is presented for calculation of photoionization rates, in fully three-dimensional grids, that improves the accuracy of the secondary processes calculation without significantly compromising the efficiency of the numerical algorithm. The method is based on generating a coarser secondary grid and interpolating the photoionization values between the two meshes, in order to overcome the enormous effort required for calculation of photoionization in gas discharge problems.A comprehensive study of the effects of photoionization, photoemission and background ionization in a short point-plane gap in air at atmospheric pressure is then presented, by using the above approach for the secondary processes in two dimensions, in conjunction with the two-dimensional axisymmetric finite-element flux-corrected transport algorithm. The secondary processes are modelled individually within a wide range of parametric values to reflect the uncertainty in the experimental data, and their effect on streamer development and propagation is investigated. The significant reduction in time required for the calculations makes numerical modelling an essential tool for better understanding of the very important yet not well understood physical processes central to the propagation and development of streamers.Finally, numerical branching is observed under certain conditions in the absence of an adequate supply of electrons in high field regions.