動量空間時空守恆法及其應用

Abstract

本論文主要的研究目的為利用高解析度動量空間時空守恆法(CESE method)模擬多種一維波動方程問題並用已知的真解當作校對，最後將之應用於強場作用下單電子原子薛丁格方程式。時空守恆法為在時間和空間上均具有二階準確度的新數值方法，此時間演化的過程是一種顯式的方法。發展動量空間時空守恆法主要的動機為避免處理複雜的非反射邊界條件於座標空間和保留完整的資訊對於散射狀態。在此我們提供了完整的推演對於非反射邊界條件的處理，亦提出了各種技巧以增加解的精確度；針對非線性問題亦提出一相對的修正；最後推導出非均勻網格之動量空間時空守恆法。由數值解與解析解比較顯示，時空守恆法模擬各種情況的波方程均具有相當精確的結果。相較於座標空間的方法，於動量空間計算波方程在時間演化下仍保持完整的資訊，對於散射狀態的問題，動量空間的方法尤其適合。

In this thesis, the purpose of investigation is to apply high accuracy numerical scheme - the "momentum space space-time conservation element and solution element (CESE) method" to simulate several one-dimensional wave equations. Several paradigmatic wave equations are solved by the method and calibrated with known solutions. Finally, we apply the method to the problem of single atom and single-active electron Schrödinger equation with strong field. The CESE method use explicit time marching and has a second-order accuracy both in space and time. Development of the CESE method in momentum space is motivated by a goal to avoid the troubles from boundary reflection, and to preserve information completely for scattering states. In this article, we complete detail deduction for the treatment of non-reflecting boundary conditions, and the skills to improve numerical accuracy. Besides, some modifications of improving numerical accuracy needed for nonlinear problems are also introduced. In the end, we derive a non-uniform grid momentum space CESE method. Comparing the numerical results with the exact solutions for each case, we have showed that the momentum CESE method produces excellent results in each kind of wave equation. Compared to the solution in coordinate space method, this method preserves the completeness of the wave’s information during the time evolution. This is a useful feature of the momentum space method especially for the scattering state problems.