论文标题
时间序列光谱法的均匀误差界限，用于具有弱非线性的非线性klein-gordon方程的长期动力学
Uniform error bounds of a time-splitting spectral method for the long-time dynamics of the nonlinear Klein-Gordon equation with weak nonlinearity
论文作者
论文摘要
我们为非线性klein-gordon方程（nkge）建立了均匀的时间误差范围（TSFP）方法，具有较弱的功率型非线性和$ o（1）$的初始数据，而非线性强度则由$ \ varepsilon^$ p} $ p. dimensionless parameter $\varepsilon \in (0, 1]$, for the long-time dynamics up to the time at $O(\varepsilon^{-β})$ with $0 \leq β\leq p$. In fact, when $0 < \varepsilon \ll 1$, the problem is equivalent to the long-time dynamics of NKGE with small initial data and $O(1)$非线性强度，而初始数据（和解决方案）的幅度为$ o（\ varepsilon）$，通过将NKGE重新构成相对论的非线性schrödinger方程，我们适应了TSFP方法，以使用数值来限制数学差异。 $ O（H^{M}+\ varepsilon^{p-β}τ^2）$ h $网格大小，$τ$ time step和$ m \ ge2 $的tsfp方法，取决于解决方案的正常范围，以适用于$ o（$ o）。 $ \ varepsilon \ in（0,1] $。尤其是，误差界限以较大时间步长的第二阶速率均匀地限制在参数$ 0 \leβ<p $。数字结果中，以确认了我们的错误范围，其差异是在参数$ 0 \ lepsilon^{ - （p-β）/2}）$中的方法。一种高度振荡的复杂NKGE，可在$ o（1）$的空间和$ o（\ varepsilon^β）$的$ O（1）$（\ Varepsilon^{ - β}）$的波速传播波长。
We establish uniform error bounds of time-splitting Fourier pseudospectral (TSFP) methods for the nonlinear Klein--Gordon equation (NKGE) with weak power-type nonlinearity and $O(1)$ initial data, while the nonlinearity strength is characterized by $\varepsilon^{p}$ with a constant $p \in \mathbb{N}^+$ and a dimensionless parameter $\varepsilon \in (0, 1]$, for the long-time dynamics up to the time at $O(\varepsilon^{-β})$ with $0 \leq β\leq p$. In fact, when $0 < \varepsilon \ll 1$, the problem is equivalent to the long-time dynamics of NKGE with small initial data and $O(1)$ nonlinearity strength, while the amplitude of the initial data (and the solution) is at $O(\varepsilon)$. By reformulating the NKGE into a relativistic nonlinear Schrödinger equation, we adapt the TSFP method to discretize it numerically. By using the method of mathematical induction to bound the numerical solution, we prove uniform error bounds at $O(h^{m}+\varepsilon^{p-β}τ^2)$ of the TSFP method with $h$ mesh size, $τ$ time step and $m\ge2$ depending on the regularity of the solution. The error bounds are uniformly accurate for the long-time simulation up to the time at $O(\varepsilon^{-β})$ and uniformly valid for $\varepsilon\in(0,1]$. Especially, the error bounds are uniformly at the second order rate for the large time step $τ= O(\varepsilon^{-(p-β)/2})$ in the parameter regime $0\leβ<p$. Numerical results are reported to confirm our error bounds in the long-time regime. Finally, the TSFP method and its error bounds are extended to a highly oscillatory complex NKGE which propagates waves with wavelength at $O(1)$ in space and $O(\varepsilon^β)$ in time and wave velocity at $O(\varepsilon^{-β})$.
