  This paper focuses on the three-dimensional simulation of the photoionization
in streamer discharges, and provides a general framework to efficiently and
accurately calculate the photoionization model using the integral form. The
simulation is based on the kernel-independent fast multipole method. The
accuracy of this method is studied quantitatively for different domains and
various pressures in comparison with other existing models based on partial
differential equations (PDEs). The comparison indicates the numerical error of
the fast multipole method is much smaller than those of other PDE-based
methods, with the reference solution given by direct numerical integration.
Such accuracy can be achieved with affordable computational cost, and its
performance in both efficiency and accuracy is quite stable for different
domains and pressures. Meanwhile, the simulation accelerated by the fast
multipole method exhibits good scalability using up to 1280 cores, which shows
its capability of three-dimensional simulations using parallel (distributed)
computing. The difference of the proposed method and other efficient
approximations are also studied in a three-dimensional dynamic problem where
two streamers interact.
