有限差分法在 Klein-Gordon-Zakharov 系统亚音速极限情形下的统一误差界
Uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime
AI总结：
本文研究了Klein-Gordon-Zakharov系统在亚音速极限情形下有限差分法的统一误差界，提出了一种均匀准确的有限差分方法，并建立了两个误差界。
摘要：
我们建立了Klein-Gordon-Zakharov系统（KGZ）中无量纲参数ε∈(0,1]的有限差分法的统一误差界，其中ε与声速成反比。在亚音速极限情形下，即ε<<1时，由于Zakharov方程中的奇异摄动或初始数据不兼容，解会传播高振荡波或快速向外扩散的初始层。具体来说，解在时间上传播O(ε)波长的波，在空间上传播O(1)波长的波，并以O(1/ε)的速度向外扩散初始层。这种高振荡时间和快速向外扩散的波对设计数值方法和建立误差估计造成显著负担。通过采用渐近一致的公式，我们提出了一种均匀准确的有限差分方法，并严格建立了两个独立的误差界，分别为O(h²+τ²/ε)和O(h²+τ+ε)，其中h是网格大小，τ是时间步长。因此，我们得到了O(h²+τ)的统一误差界，适用于0<ε≤1。主要技术包括能量方法、非线性项的截断以界数值解、振荡项的积分近似以及当ε→0⁺时KGZ解与其极限模型解之间的误差界。最后，报告了数值结果以验证我们的误差界。
英文摘要：
We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system (KGZ) with a dimensionless parameter $\varepsilon \in (0,1]$, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e. $0<\varepsilon \ll 1$, the solution propagates highly oscillatory waves in time and/or rapid outgoing initial layers in space due to the singular perturbation in the Zakharov equation and/or the incompatibility of the initial data. Specifically, the solution propagates waves with $O(\varepsilon)$-wavelength in time and $O(1)$-wavelength in space as well as outgoing initial layers in space at speed $O(1/\varepsilon)$. This high oscillation in time and rapid outgoing waves in space of the solution cause significant burdens in designing numerical methods and establishing error estimates for KGZ. By adapting an asymptotic consistent formulation, we propose a uniformly accurate finite difference method and rigorously establish two independent error bounds at $O(h^2+τ^2/\varepsilon)$ and $O(h^2+τ+\varepsilon)$ with $h$ mesh size and $τ$ time step. Thus we obtain a uniform error bound at $O(h^2+τ)$ for $0<\varepsilon\le 1$. The main techniques in the analysis include the energy method, cut-off of the nonlinearity to bound the numerical solution, the integral approximation of the oscillatory term, and $\varepsilon$-dependent error bounds between the solutions of KGZ and its limiting model when $\varepsilon\to0^+$. Finally, numerical results are reported to confirm our error bounds.
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