非线性Klein-Gordon方程在时空高振荡情况下的加权有限差分方法
Weighted finite difference methods for a nonlinear Klein-Gordon equation with high oscillations in space and time
AI总结：
针对非相对论极限下具有高度振荡初值的非线性Klein-Gordon方程，提出显式和隐式加权有限差分方法，在时空步长不受ε限制下实现二阶精度，并证明方法在ε从任意小到中等有界范围内一致收敛。
摘要：
我们考虑非相对论极限区域中的非线性Klein-Gordon方程，其初始数据为调制的高度振荡指数形式。在小尺度参数$\varepsilon\ll 1$的区域中，解在时间和空间上都表现出快速振荡。该解被近似为两个极化解的叠加，误差为$\mathcal{O}(\varepsilon)$，这些极化解是以群速度$\varepsilon^{-1}$量级反向移动的波包。极化解方程在随动坐标系中建立，然后通过显式和隐式指数加权有限差分方法进行离散。显式加权蛙跳方法需要满足CFL型稳定性条件，而隐式加权Crank-Nicolson方法无条件稳定。两种方法均达到二阶精度，且时间步长和网格尺寸不受$\varepsilon$的限制。对于极化解的近似，这些方法在$\varepsilon$从任意小到中等有界范围内一致收敛。数值实验验证了理论结果。
英文摘要：
We consider a nonlinear Klein-Gordon equation in the nonrelativistic limit regime with initial data in the form of a modulated highly oscillatory exponential. In this regime of a small scaling parameter $\varepsilon\ll 1$, the solution exhibits rapid oscillations in both time and space. The solution is approximated, up to $\mathcal{O}(\varepsilon)$, by a superposition of two polarized solutions, which are wave packets that move with opposite group velocities proportional to $\varepsilon^{-1}$. The equations for polarized solutions are formulated in co-moving coordinates and are then discretized by an explicit and an implicit exponentially weighted finite difference method. While the explicit weighted leapfrog method needs to satisfy a CFL-type stability condition, the implicit weighted Crank-Nicolson method is unconditionally stable. Both methods achieve second-order accuracy with time steps and mesh sizes that are not restricted in magnitude by $\varepsilon$. For the approximation of polarized solutions, the methods are uniformly convergent in the range from arbitrarily small to moderately bounded $\varepsilon$. Numerical experiments illustrate the theoretical results.
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