二维无条件稳定时域有限差分方法

Two-dimensional unconditionally stable FDTD method

基于半隐式的Crank Nicolson差分格式给出了一种无条件稳定时域有限差分方法。和传统FDTD法中采用的显式差分格式不同 ,对Maxwell方程组采用半隐式差分格式 ,在时间和空间上仍然是二阶精确的。但时间步长不再受稳定性条件的限制 ,只需考虑数值色散误差对其取值的制约。利用分裂场完全匹配层吸收边界截断计算空间 ,为保证PML空间的无条件稳定性 ,其方程也采用半隐式差分格式。数值结果表明相同条件下US FDTD方法与传统FDTD方法的计算精度是相同的 ,而且在增大时间步长时US FDTD方法是稳定的和收敛的。可以预见US

Based on the semi implicit Crank Nicolson difference scheme, a novel unconditionally stable 2 D finite difference time domain (US FDTD) algorithm is proposed in this paper. Different from the customary explicit difference scheme adopted in the conventional FDTD method, semi implicit difference scheme with second order accuracy in both time and space, is introduced in Maxwell equations. A remarkable advantage of the proposed method is that the Courant stability condition can be totally removed, and the time step size is limited only by the numerical dispersion errors. The split field perfectly matched layer technique is introduced to truncate computational domain, and the equations in PML medium are also differenced semi implicitly to keep unconditional stability. The numerical results from US FDTD method are consistent with that from conventional FDTD method. Additionally, with the increase of time step size, US FDTD method is stable and convergent. It can be predicated that US FDTD method is more suitable for simulation of electrically small structures, since with small space increments larger time step size can be utilized to improve computational efficiency.