Comparison of six simulation codes for positive streamers in air

We present and compare six simulation codes for positive streamer discharges from six different research groups. Four groups use a fully self-implemented code and two make use of COMSOL Multiphysics (R). Three test cases are considered, in which axisymmetric positive streamers are simulated in dry air at 1 bar and 300 K in an undervolted gap. All groups use the same fluid model with the same transport coefficients. The first test case includes a relatively high background density of electrons and ions without photoionization. When each group uses their standard grid resolution, results show considerable variation, particularly in the prediction of streamer velocities and maximal electric fields. However, for sufficiently fine grids good agreement is reached between several codes. The second test includes a lower background ionization density, and oscillations in the streamer properties, branching and numerical instabilities are observed. By using a finer grid spacing some groups were able to reach reasonable agreement in their results, without oscillations. The third test case includes photoionization, using both Luque's and Bourdon's Helmholtz approximation. The results agree reasonably well, and the numerical differences appear to be more significant than the type of Helmholtz approximation. Computing times, used hardware and numerical parameters are described for each code and test case. We provide detailed output in the supplementary data, so that other streamer codes can be compared to the results presented here.