計算複雜度

接下來本論文會對三維度LOD-FDTD 方法的計算量做個估計。由之前的文獻，吾 人可得知ADI-FDTD [30]、Split-Step FDTD [31]、以及 LOD-FDTD [34]的計算量。特別 注意上句所提到的LOD-FDTD，並沒有加上 Split-Field 的差分拆解，所以在計算複雜度 上較本論文的為少，但在推廣性上和實用性上，就是本論文為佳了。

對一個數值方法而言，討論計算複雜度的方式就是算出所有加減乘除的量。以本論 文所提的方法為例，像是(5.5) - (5.8)，(5.13) - (5.16)，(5.21) - (5.24)這幾個方程式，由於 等號左邊是被更新、被疊代的場，這些計算是在解矩陣時才會遇到的，所以我們要計算 的就是等號右邊的加減乘除總量。

對於乘和除，我們用M/D (multiplications/divisions)來標記，而加和減，則用 A/S 卻只有不分場處理的1.5 倍(36/24=1.5)。已知的事實是，LOD-FDTD 方法的計算比 ADI- FDTD 和 SS-FDTD 更迅速，這在表中也可以看得出，再加上分場的差分拆解後，LOD-

第六章 結論 (Conclusion)

本論文「二維度LOD-FDTD 方法加上 Split-Field PML 的穩定性分析」中，包括了 LOD-FDTD 方法在一維度以及二維度環境中的實現及分析。 LOD-FDTD 方法的計算。從而引入由 Berenger 所提出的 Split-Field PML 的概念，偷過 分場的技巧之後，我們重新推導一次二維度的數學差分方程，並激發高斯波模擬加以實 現，發現在邊界加上Split-Field PML 時，的確可以達到良好的吸收效果，之後更觀察了 反射係數，也驗證了這個模擬成果。

接下來我們進一步推導出其放大矩陣，發現在加了Split-Field PML 的邊界條件下，

有幾層放大矩陣的特徵值是大於1 的，這個事實說明在這個模擬情境下，此數值方法本 身的確存在著發散的行為。但是當我們再進一步去觀察更長時間的場量時，發現在二維 度的模擬中，LOD-FDTD 也能保證在經過 100’000 次的疊代計算後，也都不會出現發散 的行為。這表示其發散性雖有存在，但並不強烈，整體而言穩定性在二維度也是不錯的。

最後本論文亦提出了三維度LOD-FDTD 方法初步的推導。透過分場(Split-Field)的 拆解，成功模擬出了簡單的例子，更進一步分析複雜度，發現分場後雖處理了12 個分 量場(是不分場的兩倍)，但卻沒花到兩倍的計算量，最大的優點在於，加速 Split-Field 的實現。是非常值得繼續研究的主題。

本研究仍有許多未來的發展方向，在此我們有兩個建議。

的PML 不會導致發散，但第一層卻仍然存在導致發散危險因子。所以，或者更動電導 的函數，或者整個改變PML 的計算方式，都很可能可以得到更好的結果。

第二，嘗試實現三維度LOD-FDTD 方法加上 Split-Field PML 的模擬。對一個 FDTD 數值方法而言，如果沒有加上邊界的條件，其實的應用層面並不大。但如果以本研究為 踏板，相信可以更容易進入三維度的完整模擬，以達到實際應用的目標。但三維度的環 境下，再加上Split-Field PML 的運用，必須要考慮 12 個分量場，不管是在數學的推導 或穩定度的分析上，預期都將會是個嚴苛的挑戰。

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