Uniform error bounds of a time-splitting spectral method for the long-time dynamics of the nonlinear Klein-Gordon equation with weak nonlinearity
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We establish uniform error bounds of a time-splitting Fourier pseudospectral (TSFP) method for the nonlinear Klein-Gordon equation (NKGE) with weak cubic nonlinearity (and O(1) initial data), while the nonlinearity strength is characterized by ε^2 with ε∈ (0, 1] a dimensionless parameter, for the long-time dynamics up to the time at O(ε^-2). In fact, when 0 < ε≪ 1, the problem is equivalent to the long-time dynamics of NKGE with small initial data (and cubic nonlinearity with O(1) nonlinearity strength), while the amplitude of the initial data (and the solution) is at O(ε). By reformulating the NKGE into a relativistic nonlinear Schödinger equation (NLSE), we adapt the TSFP method to discretize it numerically. By using the method of mathematical induction to bound the numerical solution, we prove uniform error bounds at O(h^m+τ^2) of the TSFP method with h mesh size, τ time step and m>2 depending on the regularity of the solution, while the error bounds are independent of ε and they are uniformly valid for ε∈(0,1] and are uniformly accurate for the long-time simulation up to the time at O(ε^-2). Numerical results are reported to confirm our error bounds and to demonstrate that they are sharp. Finally, the TSFP method and its error bounds are extended to an oscillatory complex NKGE which propagates waves with wavelength at O(1) in space and O(ε^β) (0<β< 2) in time.
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