非线性Klein-Gordon方程长时间动力学指数波积分器的一致误差界
Uniform error bounds of an exponential wave integrator for the long-time dynamics of the nonlinear Klein-Gordon equation
摘要：
研究了一类具有三次非线性项的非线性Klein-Gordon方程（NKGE）的长时间动力学问题，建立了指数波积分傅立叶拟谱（EWI-FP）方法的一致误差界，其中三次非线性项的强度由具有无量纲参数$\varepsilon \in (0, 1]$的$\varepsilon^2$表征.当$0 < \varepsilon \ll 1$时，问题等价于具有小初始数据（和$O(1)$立方非线性）的NKGE的长时间动力学，而初始数据（和解）的振幅在$O(\varepsilon)$。对于直到$O(1/\varepsilon^{2})$的NKGE的长时间动力学，经典数值方法的分辨率和误差界显著依赖于小参数$\varepsilon$，这导致了$\varepsilon \to 0^+$的严重数值负担。EWI-FP方法是完全显式的、时间对称的，在求解波动方程时具有许多优越的性质。采用能量法与数学归纳法相结合的方法，严格地计算了EWI-FP离散在$O(h^{m_0} + \varepsilon^{2-β}τ^2)$到$O(1/\varepsilon^β)$的一致误差的界，其中$0 \leq β\leq 2$、网格尺寸$h$、时间步长$τ$和$m_0$是一个取决于解的正则性的整数。通过在时间上的重新标度，我们的结果直接扩展到误差界和$\varepsilon$ -可扩展性（或网格化策略要求）的EWI-FP方法的振荡NKGE，其解决方案传播波的波长在$O(1)$和$O(\varepsilon^β)$在空间和时间，分别和波的速度在$O(\varepsilon^{-β})$。最后，广泛的数值结果报告，以确认我们的误差估计。
英文摘要：
We establish uniform error bounds of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the nonlinear Klein-Gordon equation (NKGE) with a cubic nonlinearity whose strength is characterized by $\varepsilon^2$ with $\varepsilon \in (0, 1]$ a dimensionless parameter. When $0 < \varepsilon \ll 1$, the problem is equivalent to the long-time dynamics of the NKGE with small initial data (and $O(1)$ cubic nonlinearity), while the amplitude of the initial data (and the solution) is at $O(\varepsilon)$. For the long-time dynamics of the NKGE up to the time at $O(1/\varepsilon^{2})$, the resolution and error bounds of the classical numerical methods depend significantly on the small parameter $\varepsilon$, which causes severe numerical burdens as $\varepsilon \to 0^+$. The EWI-FP method is fully explicit, symmetric in time and has many superior properties in solving wave equations. By adapting the energy method combined with the method of mathematical induction, we rigorously carry out the uniform error bounds of the EWI-FP discretization at $O(h^{m_0} + \varepsilon^{2-β}τ^2)$ up to the time at $O(1/\varepsilon^β)$ with $0 \leq β\leq 2$, mesh size $h$, time step $τ$ and $m_0$ an integer depending on the regularity of the solution. By a rescaling in time, our results are straightforwardly extended to the error bounds and $\varepsilon$-scalability (or meshing strategy requirement) of the EWI-FP method for an oscillatory NKGE, whose solution propagates waves with wavelength at $O(1)$ and $O(\varepsilon^β)$ in space and time, respectively, and wave speed at $O(\varepsilon^{-β})$. Finally, extensive numerical results are reported to confirm our error estimates.
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