弱非线性Klein-Gordon方程四阶紧致有限差分法的长时间误差分析
Long time error analysis of the fourth-order compact finite difference methods for the nonlinear Klein-Gordon equation with weak nonlinearity
摘要：
对非线性Klein-Gordon方程（NKGE）的长时间动力学进行了四阶紧致有限差分（4cFD）离散，其中非线性强度由具有常数$p \in \mathbb{N}^+$和无量纲参数$\varepsilon \in (0, 1]$的$\varepsilon^p$表征.基于解的寿命的分析结果，在$O(\varepsilon^{-p})$上进行了4cFD方法的严格误差界。我们特别注意如何明确的误差范围取决于网格尺寸$h$和时间步长$τ$以及小参数$\varepsilon \in (0, 1]$，这表明，为了获得“正确的”数值解的时间在$O(\varepsilon^{-p})$，$\varepsilon$ -可扩展性（或网格化策略要求）的4cFD方法应采取：$h = O(\varepsilon^{p/4})$和$τ= O(\varepsilon^{p/2})$。它比经典的二阶中心差分方法具有更好的空间分辨能力。通过在时间上的重新标度，它等价于振荡NKGE，其解在空间上传播波长为$O(1)$的波，在时间上传播波长为$O(\varepsilon^p)$的波。在给定的时间内，可以直接得到振荡NKGE的误差界。最后，数值结果证实了我们的理论分析。
英文摘要：
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 $τ$ 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 $τ= O(\varepsilon^{p/2})$. 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.
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