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- W3009193670 abstract "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 $varepsilon^p$ with a constant $p in mathbb{N}^+$ and a dimensionless parameter $varepsilon in (0, 1]$. 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 $O(varepsilon^{-p})$. We pay particular attention to how error bounds depend explicitly on the mesh size $h$ and time step $tau$ as well as the small parameter $varepsilon in (0, 1]$, which indicate that, in order to obtain `correct' numerical solutions up to the time at $O(varepsilon^{-p})$, the $varepsilon$-scalability (or meshing strategy requirement) of the 4cFD methods should be taken as: $h = O(varepsilon^{p/4})$ and $tau = O(varepsilon^{p/2})$. 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 $O(varepsilon^p)$ 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." @default.
- W3009193670 created "2020-03-13" @default.
- W3009193670 creator A5039748802 @default.
- W3009193670 date "2020-03-09" @default.
- W3009193670 modified "2023-09-25" @default.
- W3009193670 title "Long time error analysis of the fourth-order compact finite difference methods for the nonlinear Klein-Gordon equation with weak nonlinearity" @default.
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- W3009193670 doi "https://doi.org/10.48550/arxiv.2003.03951" @default.
- W3009193670 hasPublicationYear "2020" @default.
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