Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

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 ε2 with 0β) 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 0β), the ε-scalability (or meshing strategy) of the FDTD methods should be taken as: h=O(εβ/2) and τ=O(εβ/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(εβ) in time. Extensive numerical results are reported to confirm our error bounds and to demonstrate that they are sharp.