22、有限差分时间域（FDTD）算法的理论与实践

有限差分时间域（FDTD）算法的理论与实践

1. 理论基础

1.1 泰勒级数收敛性

泰勒级数的收敛性取决于时间间隔 $h$ 的值以及导数的特性。对于一个正弦波形 $f(x) = \sin(\frac{2\pi x}{\lambda})$，其前六个空间导数如下：
- $\frac{\partial f(x)}{\partial x} = \frac{2\pi}{\lambda}\cos(\frac{2\pi x}{\lambda})$
- $\frac{\partial^2 f(x)}{\partial x^2} = -(\frac{2\pi}{\lambda})^2\sin(\frac{2\pi x}{\lambda})$
- $\frac{\partial^3 f(x)}{\partial x^3} = -(\frac{2\pi}{\lambda})^3\cos(\frac{2\pi x}{\lambda})$
- $\frac{\partial^4 f(x)}{\partial x^4} = (\frac{2\pi}{\lambda})^4\sin(\frac{2\pi x}{\lambda})$
- $\frac{\partial^5 f(x)}{\partial x^5} = (\frac{2\pi}{\lambda})^5\cos(\frac{2\pi x}{\lambda})$
- $\frac{\partial^6 f(x)}{\partial x^6} = -(\frac{2\pi}{\lambda})^6\sin(\frac{2\pi x}{\lambda})$

当选择点 $x = \frac{\l