在非交錯網格下發展一具最佳數值色散關係式之三維時域有限差分方法以求解馬克斯威爾方程

Abstract

本論文是在非交錯網格上提出三維時域有限差分法(FDTD)求解馬克斯威爾方程。 本文的方法是在時域內透過電場和磁場的零散度條件(亦即高斯定律)以求解法拉第定律和安培定律。 所提出的數值方法於時間離散上使用具辛結構(Symplectic)二級二階之Runge-Kutta方法，在經過長時間模擬後仍得以保持馬克思威爾方程的能量守恆性質； 另透過法拉第及安培旋度方程空間微分項的推導，得到在色散關係上相當準確的解。 本文所提出的數值方法在空間上具有四階準確，且能有效減少實解相速度與數值相速度的誤差。 本文所提出的數值方法亦顯著降低了因時域有限差分所造成的明顯地數值色散誤差以及各向異性誤差。 除了本文所做的基礎分析外，亦證實了本文所提出的數值方法在具辛結構與色散關係上具有良好的保持性，尤其在針對經長時間馬克斯威爾方程的數值模擬後，其效果尤為顯著。

An explicit finite-difference scheme for solving the three-dimensional Maxwell's equations in non-staggered grids is presented in time domain. Our aim is to solve the Faraday's and Ampere's equations in time domain within the discrete zero-divergence context for the electric and magnetic fields (or Gauss's law). The local conservation laws in Maxwell's equations are also numerically preserved all the time using the explicit second-order accurate symplectic partitioned Runge-Kutta temporal scheme. Following the method of lines, the spatial derivative terms in the semi-discretized Faraday's and Ampere's equations are then properly discretized to get a dispersively very accurate solution. This proposed fourth-order accurate space centered scheme minimizes the difference between the exact and numerical phase velocities. The significant dispersion and anisotropy errors manifested normally in finite difference time domain methods are therefore much reduced. In addition to the fundamental study performed on the proposed scheme, the dual-preserving (symplecticity and dispersion relation equation) wave solver is numerically demonstrated to be efficient for use to get in particular long-term accurate Maxwell's solutions.