We establish error bounds of the finite difference time domain (FDTD) methods for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE) with
a cubic nonlinearity, while the nonlinearity strength is characterized by ε^{2} with 0<ε≤1
a dimensionless parameter. When 0<ε≪1, it is in the weak nonlinearity regime and
the problem is equivalent to the NKGE with small initial data, while the amplitude
of the initial data (and the solution) is at O(ε). Four different FDTD methods are
adapted to discretize the problem and rigorous error bounds of the FDTD methods
are established for the long time dynamics, i.e. error bounds are valid up to the time
at O(1/ε^{β}) with 0 ≤ β ≤ 2, by using the energy method and the techniques of either
the cut-off of the nonlinearity or the mathematical induction to bound the numerical
approximate solutions. In the error bounds, we pay particular attention to how error
bounds depend explicitly on the mesh size h and time step τ as well as the small parameter ε∈(0,1], especially in the weak nonlinearity regime when 0<ε≪1. Our error
bounds indicate that, in order to get “correct” numerical solutions up to the time at
O(1/ε^{β}), the ε-scalability (or meshing strategy) of the FDTD methods should be taken
as: h=O(ε^{β}/2) and τ=O(ε^{β}/2). As a by-product, our results can indicate error bounds
and ε-scalability of the FDTD methods for the discretization of an oscillatory NKGE
which is obtained from the case of weak nonlinearity by a rescaling in time, while its
solution propagates waves with wavelength at O(1) in space and O(ε^{β}) in time. Extensive numerical results are reported to confirm our error bounds and to demonstrate
that they are sharp.