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標題: 雙重網格中央差分離散法處理在圓盤上的非線性薛丁格方程
A Two-Grid Centered Difference Discretization Scheme for the Nonlinear Schrödinger Equations on a Disk
作者: 尤力廣
Yu, Ly-Quang
關鍵字: Bose-Einstein condensates;波斯-愛因斯坦 凝聚物;time-dependent solutions;steady state solutions;two-grid centered difference method;bifurcation;disk;時間依賴解;穩定狀態解;雙重網格中央差分法;分支;圓盤
出版社: 應用數學系所
引用: References Abramowitz, M. & Stegun, I. A. [1964] Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC). Adhikari, S. K. [2000] “Numerical study of the spherically symmetric Gross-Pitaevskii equation in two space dimensions,” Phys. Rev. E 62 2937-2944. Agrawal, G. P. [2001] Nonlinear Fiber Optics, 3rd edn. (Academic Press, New York). Akhmediev, N. & Ankiewice, A. [1999] “Partially coherent solitons on a finite fackground,” Phys. Rev. Lett. 82, 2661-2664. Allgower, E. L. & Georg, K. [2003] Introduction to Numerical Continuation Methods (SIAM Publications, Philadelphia). Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E. & Cornell, E. A. [1995] “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198-201. Anglin, J. R. & Ketterle, W. [2002] “Bose-Einstein condensation of atomic gases,” Nature 416, 211-218. Bao, W. & Du, Q. [2004] “Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow,” SIAM J. Sci. Comput. 25, 1674-1697. Bao, W., Jacksch, D. & Markowich, P. A. [2003] “Numerical solution of Gross-Pitaevskii equation for Bose-Einstein condensation,” J. Comput. Phys. 187, 318-342. Bao, W., Jin, S. & Markowich, P. A. [2002] “On time splitting spectral approximations for the Schr¨odinger equation in the semiclassical regime,” J. Comput. Phys. 175, 487-524. Bao, W., Jin, S. & Markowich, P. A. [2003] “Numerical study of time-splitting spectral discretizations of nonlinear Schr¨odinger equations in the semiclassical regimes,” SIAM J. Sci. Comput. 25, 27-64. Bao, W. & Tang, W. [2003] “Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional,” J. Comput. Phys. 187, 230-254. Bradley, C. C., Sackett, C. A. & Hulet, R. G. [1997] “Bose-Einstein condensation of Lithium: observation of limited condensate number,” Phys. Rev. Lett. 78, 985-989. Chang, S.-L. & Chien, C.-S. [2006] “Liapunov-Schmidt reduction and continuation for the coupled nonlinear Schr¨odinger equation,” submitted to SIAM J. Sci. Comput.. Chang, S.-L. & Chien, C.-S. [2007] “Numerical continuation for nonlinear Schr¨odinger equations,” Inter. J. Bifurcation and Chaos, to appear. Chang, S.-L., Chien, C.-S. & Jeng, B.-W. [2005] “Tracing the solution surface with folds of a two-parameter system,” Inter. J. Bifurcation and Chaos 15, 2689-2700 Chien, C.-S. & Jeng, B.-W. [2006] “A two-grid discretization scheme for semilinear elliptic eigenvalue problems,” SIAM J. Sci. Comput. 27, 1287-1304. Gobulitsky M., Stewart, I. & Schaeffer, D.G. [1988] Singularities and Groups in Bifurcation Theory, Vol. II (Springer-Verlag, New York). Haelterman, M. & Sheppard, A. [1994] “Bifurcation phenomena and multiple soliton-bound states in isotropic Kerr media,” Phys. Rev. E 49, 3376-3381. Keller, H. B. [1987] Lectures on Numerical Methods in Bifurcation Problems (Springer-Verlag, Berlin). Kuttler, J. R. & Sigillito, V. G. [1984] “Eigenvalues of the Laplacian in two dimensions,” SIAM Rev. 26, 163-193. Lai, M.-C. [2001] “A note on finite difference discretizations for Poisson equation on a disk,” Numer Methods Partial Differential Eq. 17, 199-203. Lin, T.-C. & Wei, J. [2005] “Ground state of N coupled nonlinear Schr¨odinger equations in Rn, n 3,” Comm. Math. Phys. 255, 629-653. Xu, J. [1994] “A novel two-grid method for semilinear elliptic equations,” SIAM J. Sci. Comput. 15, 231-237.
我們提出一個創新的演算法求解M個非線性薛丁格方程 (MCNLS)與時間有關的波函數。首先,我們使用變數分離法將MCNLS轉換為穩定狀態的方程。我們討論定義在圓盤上線性特徵值問題的線性穩定。我們描述一個具有雙重迴圈延續法的雙重網格中央差分離散法來計算MCNLS的穩定狀態解。此處,我們將化學位能當作為延續參數,並利用預測修正延續法來追蹤MCNLS的解曲線和曲面。當我們得到數值解流型後,則波函數可以很容易得到。我們所提出的演算法具有不必離散或積分波函數偏導數的優點。此外,在任何時間尺度下,波函數也可以被得到。我們的數值試驗則是考慮定義在單位圓盤上的MCNLS問題。並提出一些數值結果的報告。

We present a novel algorithm for computing time-dependent solutions of M-coupled nonlinear Schrödinger equations (MCNLS). First we transform the MCNLS equations to the steady state equations by using separation of variables. Then we discuss the linear stability of the linear eigenvalue problem defined on a disk. A two-grid centered difference discretization scheme with two-loop continuation algorithms for computing steady state solutions of the MCNLS is described. The chemical potentials are treated as continuation parameters. A predictor-corrector continuation method is exploited to trace solution curves and surfaces of the MCNLS. The time-dependent solutions can be easily obtained whenever the solution manifolds are numerically traced. The proposed algorithm has the advantage that it is unnecessary to discretize or integrate the partial derivatives of wave functions. Moreover, the wave functions can be evaluated for any time scale. Our numerical computations were executed on a unit disk. Sample numerical results are reported.
其他識別: U0005-2406200617223400
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