论文标题
对于非线性klein-gordon方程的四阶紧凑型差异方法的长期错误分析，非线性弱
Long time error analysis of the fourth-order compact finite difference methods for the nonlinear Klein-Gordon equation with weak nonlinearity
论文作者
论文摘要
我们介绍了非线性klein-gordon方程（NKGE）的长时间动态的四阶有限差（4CFD），而非线性强度则以$ \ varepsilon^p $为特征，其常数$ p \ in \ in \ n}^+$ $ a {n}^+$ $ parep $ parte $ p \ in（n}解决方案寿命的分析结果，4CFD方法的严格误差界限是在$ o（\ varepsilon^{ - p}）$时进行的。 ``正确的''in $ o（\ varepsilon^{ - p}）$的数值解决方案，4cfd方法的$ \ varepsilon $ -scalibility（或Meshight策略要求）应为：$ h = o（\ h = o（\ v varepsilon^varepsilon^{p/4}）与经典的二阶中心差异方法相比，它的空间分辨率更好。 分析。
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.
