论文标题
非线性klein-gordon方程的长期动力学的指数波积分器的均匀误差界限
Uniform error bounds of an exponential wave integrator for the long-time dynamics of the nonlinear Klein-Gordon equation
论文作者
论文摘要
我们建立了指数波积分器的均匀误差界傅立叶伪谱（EWI-FP）方法，用于非线性klein-gordon方程（nkge）的长期动态，其强度的特征在于$ \ varepsilon^2 $，其强度为$ \ varepsilon f in（varepsilon f in（varepsilon），$ \ varepsilon f in（varepsilon^2 $ in（0） \ ll 1 $，问题与NKGE的长期动态相当于少量的初始数据（以及$ o（1）$ cubic非线性），而初始数据的幅度（和解决方案）在$ O（\ varepsilon）$（\ varepsilon）$（$ o（\ VAREPSILON）$）的长时间（$ o（1/of）的长期动态时。经典的数值方法显着取决于小参数$ \ varepsilon $，这会导致严重的数值负担为$ \ varepsilon \至0^+$。 EWI-FP在$ O（H^{M_0} + \ varepsilon^{2-β}τ^2）$（1/\ VAREPSILON^β）$的时间为$（1/\ varepsilon^β）$，$ 0 \ leq leqβ\ leqβ\ leq 2 $，网格尺寸$ h $，时间$ h $，time $ h $ fress $τ$τ$τ$ m_0随着时间的流逝，我们的结果直接扩展到误差界限和$ \ varepsilon $ -Scalability（或Meshing策略要求）的EWI-FP方法，用于振荡性NKGE，其解决方案的解决方案在$ o（1）$ o（1）$和$ o（\ varepsilon^β）$（\ varepsilon^β）上的波长传播，并在太空中及时相应，并相应地相差$ o（\ varepsilon^{ - β}）$。
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.
