Long time error analysis of the fourth-order compact finite difference methods for the nonlinear Klein-Gordon equation with weak nonlinearity
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We present the fourth-order compact finite difference (4cFD) discretizations for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE), while the nonlinearity strength is characterized by ε^p with a constant p ∈N^+ and a dimensionless parameter ε∈ (0, 1]. Based on analytical results of the life-span of the solution, rigorous error bounds of the 4cFD methods are carried out up to the time at O(ε^-p). 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], which indicate that, in order to obtain `correct' numerical solutions up to the time at O(ε^-p), the ε-scalability (or meshing strategy requirement) of the 4cFD methods should be taken as: h = O(ε^p/4) and τ = O(ε^p/2). It has better spatial resolution capacity than the classical second order central difference methods. By a rescaling in time, it is equivalent to an oscillatory NKGE whose solution propagates waves with wavelength at O(1) in space and O(ε^p) in time. It is straightforward to get the error bounds of the oscillatory NKGE in the fixed time. Finally, numerical results are provided to confirm our theoretical analysis.
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