高效能周道積分法模擬表面電漿特徵值問題

Abstract

數值模擬是研究表面電漿特性的重要方法。本文以馬克士威方程式建模，將此方程用 K.S.Yee 提出的時域有限差分法進行離散化 (discretization)，接著將離散後的方程做適當的相似變換 (similarity transformation)，使原始的問題轉換為特徵值問題 (eigenvalue problem)。此特徵值問題的矩陣為非共軛對稱 (non-Hermitian) 的方陣，並且其特徵值分佈呈現高度叢集性。因此為克服現有方法在解此問題所需特定區域特徵值的困難，而發展了一個高效的周道積分法應用於求解此問題。此方法結合了周道積分、快速矩陣向量乘法以及高效的線性系統求解。由數據結果可驗證本文提出之方法能高效求解線性系統以及特徵值。

Numerical simulations play a significant role for studying the properties of surface plasmon. The surface plasmon problem is first modelled by the Maxwell equations, and the equations is then discretized by the widely-used Yee’s scheme. After applying certain similarity transformations to the discretized system, the original simulation problem becomes a clustered non-Hermitian eigenvalue problem. An efficient contour integral (CI) based eigensolver is developed to overcome the difficulties of applying current existing methods to solve eigenvalues in particular designated regions for this problem. This efficient method combines the contour integral, the fast matrix-vector multiplication and efficient linear system solving. The numerical results can show the efficiency of solving linear systems and eigenvalues with the efficient CI eigensolver.