具辛結構及頻散保持性質之平行馬克斯威爾方程算則

On a symplecticity and dispersion relation preserving parallel solver for Maxwell's equations

蔡明宏(Ming-Hong Tsai)

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(2009年)

https://doi.org/10.6342/NTU.2009.02458

摘要

對於光電電磁波，為了能夠解決奇、偶震盪的問題並得以節省計算時間，本論文中在空間內使用一種準確且有效率得以保持光波傳遞頻散關係Dispersion- relation -preserving (DRP) 的非交錯有限差分格式來離散一次微分項。針對長時間之數值模擬，為了保持其漢米爾頓的結構，使用一有效率守恆物理量的具辛 (Symplectic) 性質的差分格式，並將此一Symplecitic DRP的差分方法其運用於離散光電磁學方程Maxwell’s equations，以求解空間中包含散射體之電磁問題。
藉由求解二維及三維的光電電磁波方程，證實本論文所提出之求解程序的準確性及可行性，由測試問題可知，本論文所提出之格式，在所有的測試問題中均能保有相當好的收斂斜率及能量守恆性。
為了模擬無限域問題，本文中使用了完美匹配層(PML)、全場/散射場(TF/SF)與等位函數法(Level Set)等數值技巧，求解包含非均勻介質之電磁問題(包括二維TM模態米氏電磁散射問題和三維米氏電磁散射問題，以及二維TM模態複雜非均勻介質光子晶體波導問題)，經由測試題目可以得知，本論文所提出的方法可以得到相當好的準確性，且與前人所模擬之結果均呈相當的吻合性。
最後，針對三維米氏電磁散射問題，本文使用叢集式電腦及訊息溝通介面 (message passing interface MPI) 函式庫，將序列程式平行化，藉由使用區域分割法及本論文所提出的平行差分格式，本論文均求得不錯的平行加速比及效能比的結果。

關鍵字

馬克斯威爾方程 ； 頻散保持 ； 辛結構

並列摘要

In this thesis, the electromagnetic wave equation
is discretized in non-staggered grids.
To avoid even-odd spurious oscillations,
the first-order spatial derivative terms will be approximated
by the explicit compact scheme to save the computational time.
To accommodate the Hamiltonian structure in the Maxwell's equations,
the time integrator employed in the current semi-discretization
falls into the symplectic category.
The integrity of the finite difference time domain method for solving
the Maxwell's equations involving scatters will be verified by solving
several problems in two- and three-dimensional that are amenable to the exact solutions.
The results with good rates of convergence are demonstrated
for all the investigated problems.
For simulating wave problems on open domain, in this thesis, the Perfectly matched layer (PML), Total-field-Scattered-field (TF/SF) and Level Set method are employed for solving scattering problems, including 2-D (TM) Mie scattering problem, 3-D Mie scattering problem and
modeling of PC-based L-shaped waveguide problem. The results simulated from the proposed method agree well with other numerical and experimental results for the chosen problems.
Finally, the present Maxwell's equation solver for the 3-D Mie scattering problem
are solved in MPI parallel platforms.
With the domain decomposition methods combined with the proposed
scheme, the speed-up and efficiency are both good in the simulated scattering problem.

並列關鍵字

dispersion relation preserving ； symplecticity

參考文獻

[1] Y. Liu, Fourier analysis of numerical algorithms for the Maxwell’s equations, J.

Comput. Phys. 124 (1996) 396-416.

[2] C. W. Trowbridge, J. K. Sykuski, Some key developments in computational electromagnetics

and their attribution, IEEE Trans. Magn. 42 (2006) 503-508.

Finite Element Solutions to Time Dependent Maxwell Equations, IEEE Trans. Antenn.