發展一具辛結構及最佳數值色散關係式的方法以求解具色散物質的馬克斯威爾方程

Development of a symplectic scheme with optimized numerical dispersion-relation equation to solve Maxwell's equations in dispersive media

張日陽(Rih-Yang Chung)

指導教授 ： 許文翰

國立臺灣大學/工學院/工程科學及海洋工程學研究所/碩士(2012年)

https://doi.org/10.6342/NTU.2012.01542

摘要

本論文是在交錯網格上發展一具最佳數值色散關係之馬克斯威爾方程組的時域有限差分法。本文的目的是在進行長時間的模擬時，於非色散性介值空間仍能保持電場與磁場的零散度條件及維持其能量守恆之性質于時間離散使用了具辛結構(Symplectic) 二級二階之Runge-Kutta；而在半離散式之法拉第及安培旋度方程的空間微分項上，依數值角頻率與波數之間相依的觀念，引入了色散關係方程(Dispersion relation equation, DRE)，以期在空間離散上具最佳數質色散的空間微分近似。為了達到最佳數值色散，我們提出減少實解與數值解色散關係方程之誤差的方式，並在空間上得到四階準確之中央差分離散格式。最後，我們把重點放在三種典型代表性之色散介質Debye、Lornetz、Drude 介質模型的電磁波模擬。透過計算上的運用，證明了文中所發展之數值方法於馬克斯威爾方程組在與頻率獨立和頻率相依上的有效性及在長時間模擬下的準確度。

關鍵字

馬克斯威爾方程 ； Debye 介質 ； Lorentz 介質 ； Drude 介質 ； 色散關 係式 ； 顯式具辛結構 ； 頻率相依

並列摘要

In this paper an explicit nite-di erence scheme is developed in staggered grids for solving the Maxwell's equations in time domain. It is aimed to preserve the discrete zero-divergence condition in the electrical and magnetic
elds and conserve some inherent laws in non-dispersive media all the time using the explicit second-order accurate symplectic partitioned Runge-Kutta temporal scheme. The spatial derivative terms in the resulting semi-discretized
Faraday's and Ampere's equations are approximated to get an accurate mathematical dispersion relation equation that governs the numerical angular frequency and the wavenumber for the Maxwell's equations in a domain of two space dimensions. To achieve the goal of getting the best dispersive characteristics, a fourth-order accurate space centered scheme with the ability of minimizing the di erence between the exact and numerical dispersion relation equations is proposed. The emphasis of this study is placed on the accurate modelling of EM waves in the dispersive media of the Debye, Lorentz and Drude types. Through the computational exercises, the proposed dualpreserving solver is computationally demonstrated to be e cient for use to
predict the long-term accurate Maxwell's solutions for the media of frequency independent and dependent types.

並列關鍵字

Debye ； Lorentz ； Drude ； dual-preserving solver ； dispersion relation equation ； frequency dependent

參考文獻

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