  In this paper, we propose an efficient and accurate message-passing interface
(MPI)-based parallel simulator for streamer discharges in three dimensions
using the fluid model. First, we propose a new second-order semi-implicit
scheme for the temporal discretization of the model that relaxes the dielectric
relaxation time restriction. Moreover, it requires solving the Poisson-type
equation only once at each time step, while the classical second-order explicit
scheme typically needs to do twice. Second, we introduce a geometric multigrid
preconditioned FGMRES solver that dramatically improves the efficiency of
solving the Poisson-type equation with either constant or variable
coefficients. We show numerically that no more than 4 iterations are required
for the Poisson solver to converge to a relative residual of $10^{-8}$ during
streamer simulations; the FGMRES solver is much faster than R&B SOR and other
Krylov subspace solvers. Last but not least, all the methods are implemented
using MPI. The parallel efficiency of the code and the fast algorithmic
performances are demonstrated by a series of numerical experiments using up to
2560 cores on the Tianhe2-JK clusters. For applications, we study a
double-headed streamer discharge as well as the interaction between two
streamers, using up to 10.7 billion mesh cells.
