以半顯隱Runge-Kutta及擬譜法求解薛丁格方程

Implicit-Explicit Runge-Kutta and pseudospectral methods for Schrodinger equation

林沛沅(Pei-Yuan Lin)

指導教授 ： 賴明治 ； 鄧君豪

國立交通大學/理學院/數學建模與科學計算研究所/碩士(2014年)

https://doi.org/10.6842/NCTU.2014.00001

摘要

本論文使用半顯隱Runge-Kutta和擬譜法建立計算格式以求解薛丁格方程。利用補償法將邊界條件加入格式中，藉由離散的能量估計，訂立適當的懲罰參數。我們應用Legendre-Gauss-Lobatto 及Chebyshev-Gauss-Lobatto這兩組不同的網格點進行計算，並從幾個數值實驗來驗證此格式。

關鍵字

擬譜法 ； 薛丁格方程 ； 補償法 ； 半顯隱Runge-Kutta

並列摘要

In this paper, we present a scheme for solving the Schrödinger equation based on Implicit-Explicit Runge-Kutta and pseudospectral method. The boundary conditions are imposed to the scheme through the penalty methodology. By conducting the energy estimate, we determine the values of penalty parameters. We apply Legendre-Gauss-Lobatto and Chebyshev-Gauss-Lobatto grid points for numerical computations. Several numerical experiments are shown to validate the scheme.

並列關鍵字

pseudospectral method ； Schrodinger equation ； penalty method ； Implicit-Explicit Runge-Kutta

參考文獻

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methods on Chebyshev points. SIAM J. Numer. Anal., 31 (1994), pp. 1519-1534.

[3] C.A. Kennedy, M.H. Carpenter, Additive Runge-Kutta schemes for convectiondi

[4] P. D. Lax, R. D. Richtmyer, Survey of the stability of linear nite dierence equations.