  Particle-in-Cell (PIC) simulation is the most important numerical tool in
plasma physics. However, its long-term accuracy has not been established. To
overcome this difficulty, we developed a canonical symplectic PIC method for
the Vlasov-Maxwell system by discretizing its canonical Poisson bracket. A fast
local algorithm to solve the symplectic implicit time advance is discovered
without root searching or global matrix inversion, enabling applications of the
proposed method to very large-scale plasma simulations with many, e.g.,
$10^{9}$, degrees of freedom. The long-term accuracy and fidelity of the
algorithm enables us to numerically confirm Mouhot and Villani's theory and
conjecture on nonlinear Landau damping over several orders of magnitude using
the PIC method, and to calculate the nonlinear evolution of the reflectivity
during the mode conversion process from extraordinary waves to Bernstein waves.
