臺灣博碩士論文加值系統

在這篇論文裡,我們研究裁縫有限點法與預測修正延續法來解一些特徵值問 題,包括簡單線性特徵值問題、非線性特徵值問題、線性薛丁格特徵值問題、非 線性的薛丁格特徵值問題和耦合非線性薛丁格特徵值問題。首先我們利用裁縫有 限點法離散上述的特徵值問題,接著利用預測修正延續法追蹤解的曲線。由我們 的實驗數值測試結果顯示,對於有解析解的問題,其誤差的收斂為 O(h︿2)。

We study a Tailored Finite Point Method (TFPM) and predictor-corrector continuation method for solving eigenvalue problems include simple linear eigenvalue problem, nonlinear eigenvalue problem, linear Schrodinger eigenvalue problem, nonlinear Schrodinger eigenvalue problem, and coupled nonlinear Schrodinger eigenvalue problem. First, we present a tailored finite point method to discretize above equa- tions, and trace the solution curve by predictor-corrector continuation method. The numerical results of the problem with exact solution show that error for TFPM is in order of O(h︿2).

1. Introduction 1
1.1. History review............................... 1
1.2. Outline ........................................ 2

2. Model Problem 3
2.1. Schrodinger equation........................... 3

3. Tailored finite point method 5
3.1. TFPM for simple linear eigenvalue problem............................ 5
3.2. TFPM for the linear Schrodinger eigenvalue problem..........9
3.3. TFPM for nonlinear Schrodinger eigenvalue problem..........11

4. A brief review of Continuation Method 13
4.1. Introduction.............................................................................. 13
4.2. The predictor-corrector continuation method............................ 13
4.3. The continuation method for linear eigenvalue problem..............16
4.4. The continuation method for coupled nonlinear eigenvalue problem ............................................................................................................17

5. Numerical Exmples 19
5.1. Example 1: The simple linear eigenvalue problem........................19
5.2. Example 2: The nonlinear eigenvalue problem............................20
5.3. Example 3: The linear Schrodinger eigenvalue problem.............. 20
5.4. Example 4: The nonlinear Schrodinger eigenvalue problem..........20
5.5. Example 5: The coupled of nonlinear Schrodinger eigenvalue problem .............................................................................................21

6. Conclusions 22

[1] M. Abramowitz and I.A. Stegun, “Handbook of Mathematical Functions, National Bureau of Standards, 1964.

[2] E.L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, SIAM Publications, Philadelphia, 2003.

[3] S.L. Chang, C.S. Chien, “Adaptive continuation algorithms for computing en- ergy levels of rotating Bose-Einstein condensate, Comput. Phys. Commun. (2007) 177: 707-719.

[4] D. C. Dzeng and W.W. Lin, “Homotopy continuation method for the numerical solutions of generalised symmetric eigenvalue problems, J. Austral. Math. Soc. Ser. (1991) 32: 437-456.

[5] H. Han, Z. Huang, and R.B. Kellogg, “A tailored finite point method and a problem of P. Hemker, Proceedings of the International Conference on Boundary and Interior Layers - Computational and Asymptotic Methods, Limerick, July 2008.

[6] H. Han, Z. Huang, and R.B. Kellogg, “A tailored finite point method for a singular perturbation problem on an unbounded domain, J. Sci. Comput. (2008) 36: 243-261.

[7] H. Han and Z. Huang, “Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions, J. Sci. Comput., (2009) 41: 200-220.

[8] H. Han and Z. Huang, “A tailored finite point method for the Helmholtz equation with high wave numbers in heterogeneous mediun, J. Sci. Cumput. Math. (2008) 26: 728-739

[9] Hans D. Mittelmann, “A Pseudo-Arclength Continuation Method for nonlinear eigenvalue problems, SIAM. J. Numer. Anal. (1986) 23:1007-1016.

[10] Y. Shih, R.B. Kellogg, and P. Tsai, “A Tailored Finite Point Method for Convection-Diffusion-Reaction Problems, J. Sci. Comput. (2010) 43: 239-260.

[11] Y. Shih, R.B. Kellogg, and Y. Chang, “Characteriatic Tailored Finite Point Method for Convection-Dominated Convection-Diffusion-Reaction Problems, J. Sci. Compu. (2011) 47: 198-215.

[12] A. Tveito and R. Wathen, Introduction to Partial Differential Equations, Springer-Verlag, New York, 1998.