Schr（？）dinger方程的数值解法
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摘要
本文致力于数值求解Schrodinger方程的差分方法的研究,主要包含针对一维Schrodinger方程的Obrechkoff方法和针对含时Schrodinger方程的时间空间离散方法两部分内容。
第一部分是应用Obrechkoff方法数值求解一维Schrodinger方程,在第二章中研究了用Obrechkoff单步法来离散一维Schrodinger方程,利用Mathematica的多精度计算软件包来求得任意精度的数值解,着重探讨了指数拟合方法对于求解Schrodinger方程的束缚态和共振态本征值的精度的影响,数值实验表明对于高能级的共振态,指数拟合Obrechkoff单步法比不拟合的情形在精度和效率上都有很大的提高。第三章中本文提出一种基于Obrechkoff单步法组合的P稳定两步法,其特点是在差分格式中,加入连续高阶微商,而这种结构仍能保持P稳定性,同M. Van Daele和G. Vanden Berghe的方法相比,当两种方法在使用相同的最高阶的微商时,本文提出的P稳定两步法在精度上很大地超越了他们的方法。在第四章中,本文继续第三章关于单步法组合成为两步法的讨论,得到了更高精度的两步方法,进而对其稳定性进行了研究,提出了一种普遍的适用于两步法的相位延迟(phase-lag)公式,对这种新的两步格式进行三角函数拟合得到了相位延迟阶数为无穷的三角函数拟合P稳定两步法,本文应用这种两步法求解Schrodinger方程的数值解,显示出它的精度和效率的优越性。
在第二部分,本文关注含时Schrodinger方程的数值解法。文中改进了其他小组特别是W. van Dijk和F. M. Toyama对于含时Schrodinger方程的空间离散方法,在他们的结构上充分加入离散各点波函数的两阶微商,从而将方法的精度从O(h2l)提高到O(h4l),同时采用Pade近似来计算时间演化算符,从而在时间演化算符计算方面可以达到相当高的精度。本文用LU分解来求解对时空离散后得到的含(2l+1)对角矩阵的矩阵方程,从而有效的得到高精度的数值解。
This dissertation is devoted to numerical methods for the Schrodinger equation. The following two parts are included:
The first is about how to overcome the barrier for a finite difference method of the one-dimensional Schroinger equation defined on the infinite integration interval to obtain the nu-merical solutions accurate than the standard precision in most computing systems of 10-16. In chapter 2, the Obrechkoff one-step method implemented in the multi precision mode is employed to obtain the numerical solutions of the Woods-Saxon potential with errors less than 10-50 and 10-30 for the bound and resonant state, respectively, within a reasonable ef-ficiency. In this dissertation, I also investigate that how to use exponentially-fitting to im-prove the Obrechkoff one-step method in finding the numerical solutions of both the bound and resonant states of the Schroinger equation. The numerical experiments show that the exponentially-fitted method, when the number of fitted coefficients is not so much, can im-prove the precision of the eigenvalues in the bound and resonant state in the lower-order case, nevertheless the advantage in the precision is not so remarkable in the higher-order case. If the coefficients of the Obrechkoff one-step method are full fitted by the exponential function as shown in our work, these methods will surpass the non-fitted Obrechkoff one-step method in accuracy and efficiency considerably for finding out the numerical solutions of the high level resonant states. In chapter 3, a new P-stable Obrechkoff two-step method, which is based on the Obrechkoff one-step method, is presented. Compared to the scheme of M. Van Daele and G. Vanden Berghe, the present one, which contains the high-order derivatives of both even and odd order, can greatly reduce the error. Five numerical examples, which includes the Woods-Saxon potential, the Morse potential, the modified Poschl-Teller potential, the Stiefel-Betis problem, and the Duffing equation, are given to illustrate the performance of this method. Chapter 4 concerns the trigonometrically-fitted two-step method with multi-derivative for the numerical solution to the one-dimensional Shrodinger equation. In this chapter, a general formula of the Phase-lag for the Obrechkoff two-step method is presented.
The second is for accurate numerical solutions of the time-dependent Schrodinger equa-tion (TDSE). We present an improved space discretization scheme for the numerical solutions of the TDSE. Compared to the scheme of van Dijk and Toyama,the present one, which con- tains more terms of second-order partial derivatives, can greatly reduce the error resulting from the integration over the space. For a (2l+1)-point formula with (2l+1) terms of second-order partial derivatives, the local truncation error can decrease from the order of (△x)2l to (△x)4l, while the previous one contains only one term of second-order partial derivative. In addition, we employ the high-order Pade approximant for the time evolution operator. Two well-known numerical examples and the corresponding error analysis demonstrate that the present scheme has the advantage in the precision and efficiency over the previous one.
第一部分是应用Obrechkoff方法数值求解一维Schrodinger方程,在第二章中研究了用Obrechkoff单步法来离散一维Schrodinger方程,利用Mathematica的多精度计算软件包来求得任意精度的数值解,着重探讨了指数拟合方法对于求解Schrodinger方程的束缚态和共振态本征值的精度的影响,数值实验表明对于高能级的共振态,指数拟合Obrechkoff单步法比不拟合的情形在精度和效率上都有很大的提高。第三章中本文提出一种基于Obrechkoff单步法组合的P稳定两步法,其特点是在差分格式中,加入连续高阶微商,而这种结构仍能保持P稳定性,同M. Van Daele和G. Vanden Berghe的方法相比,当两种方法在使用相同的最高阶的微商时,本文提出的P稳定两步法在精度上很大地超越了他们的方法。在第四章中,本文继续第三章关于单步法组合成为两步法的讨论,得到了更高精度的两步方法,进而对其稳定性进行了研究,提出了一种普遍的适用于两步法的相位延迟(phase-lag)公式,对这种新的两步格式进行三角函数拟合得到了相位延迟阶数为无穷的三角函数拟合P稳定两步法,本文应用这种两步法求解Schrodinger方程的数值解,显示出它的精度和效率的优越性。
在第二部分,本文关注含时Schrodinger方程的数值解法。文中改进了其他小组特别是W. van Dijk和F. M. Toyama对于含时Schrodinger方程的空间离散方法,在他们的结构上充分加入离散各点波函数的两阶微商,从而将方法的精度从O(h2l)提高到O(h4l),同时采用Pade近似来计算时间演化算符,从而在时间演化算符计算方面可以达到相当高的精度。本文用LU分解来求解对时空离散后得到的含(2l+1)对角矩阵的矩阵方程,从而有效的得到高精度的数值解。
This dissertation is devoted to numerical methods for the Schrodinger equation. The following two parts are included:
The first is about how to overcome the barrier for a finite difference method of the one-dimensional Schroinger equation defined on the infinite integration interval to obtain the nu-merical solutions accurate than the standard precision in most computing systems of 10-16. In chapter 2, the Obrechkoff one-step method implemented in the multi precision mode is employed to obtain the numerical solutions of the Woods-Saxon potential with errors less than 10-50 and 10-30 for the bound and resonant state, respectively, within a reasonable ef-ficiency. In this dissertation, I also investigate that how to use exponentially-fitting to im-prove the Obrechkoff one-step method in finding the numerical solutions of both the bound and resonant states of the Schroinger equation. The numerical experiments show that the exponentially-fitted method, when the number of fitted coefficients is not so much, can im-prove the precision of the eigenvalues in the bound and resonant state in the lower-order case, nevertheless the advantage in the precision is not so remarkable in the higher-order case. If the coefficients of the Obrechkoff one-step method are full fitted by the exponential function as shown in our work, these methods will surpass the non-fitted Obrechkoff one-step method in accuracy and efficiency considerably for finding out the numerical solutions of the high level resonant states. In chapter 3, a new P-stable Obrechkoff two-step method, which is based on the Obrechkoff one-step method, is presented. Compared to the scheme of M. Van Daele and G. Vanden Berghe, the present one, which contains the high-order derivatives of both even and odd order, can greatly reduce the error. Five numerical examples, which includes the Woods-Saxon potential, the Morse potential, the modified Poschl-Teller potential, the Stiefel-Betis problem, and the Duffing equation, are given to illustrate the performance of this method. Chapter 4 concerns the trigonometrically-fitted two-step method with multi-derivative for the numerical solution to the one-dimensional Shrodinger equation. In this chapter, a general formula of the Phase-lag for the Obrechkoff two-step method is presented.
The second is for accurate numerical solutions of the time-dependent Schrodinger equa-tion (TDSE). We present an improved space discretization scheme for the numerical solutions of the TDSE. Compared to the scheme of van Dijk and Toyama,the present one, which con- tains more terms of second-order partial derivatives, can greatly reduce the error resulting from the integration over the space. For a (2l+1)-point formula with (2l+1) terms of second-order partial derivatives, the local truncation error can decrease from the order of (△x)2l to (△x)4l, while the previous one contains only one term of second-order partial derivative. In addition, we employ the high-order Pade approximant for the time evolution operator. Two well-known numerical examples and the corresponding error analysis demonstrate that the present scheme has the advantage in the precision and efficiency over the previous one.
引文
[1]徐龙道等编著,物理学词典[M],(科学出版社,2004),pp.847.1
[2]J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Vol.1: The Quantum Theory of Planck, Einstein, Bohr and Sommerfeld. Its Foundation and the Rise of Its Difficulties (1900-1925)[M]. (New York:Springer-Verlag,1982).1
[3]M. Planck, On the Law of Distribution of Energy in the Normal Spectrum[J], Ann. Physik, vol.4,1901, pp.553 ff.1
[4]A. Einstein, On a Heuristic Viewpoint Concerning the Production and Transformation of Light[J], Ann. Physik, vol.17,1905, pp.132-148.1
[5]曾谨言著,量子力学卷二[M],(科学出版社,1997),pp.1-2.1
[6]W. Heisenberg, Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen[J], Z. f. Physik, vol.33,1925, pp.879-893.1
[7]M. Born and P. Jordan, Zur Quantenmechanik[J], Z. f. Physik, vol.34,1925, pp.858-888.
[8]M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik Ⅱ[J], Z. f. Physik, vol. 35,1925, pp.557-615.1
[9]L. de Broglie, Recherches sur la theorie des quanta (Researches on the quantum the-ory)[D], Thesis, (Paris,1924).1
[10]E. Schrodinger, An Undulatory Theory of the Mechanics of Atoms and Molecules[J], Phys. Rev., vol.28,1926, pp.1049-1070.1
[11]J. Mehra, Erwin Schrodinger and the Rise of Wave Mechanics. Ⅰ. Schrodinger's Scien-tific Work before the Creation of Wave Mechanics[J], Foundations of Physics, vol.17, 1987, pp.1051-1112.
[12]J. Mehra, Erwin Schrodinger and the Rise of Wave Mechanics. Ⅱ. The Creation of Wave Mechanics[J], Foundations of Physics, vol.17,1987, pp.1141-1188.
[13]J. Mehra, Erwin Schrodinger and the Rise of Wave Mechanics. Ⅲ. Early Response and Applications[J], Foundations of Physics, vol.18,1988, pp.107-184.1
[14]P. A. M. Dirac, Ph.D. Thesis[D]. (Cambridge University,1926).1
[15]M. Born, Zur Quantenmechanik der Stoβvorgange[J], Z. f. Physik, vol.37,1926, pp. 863-867.1
[16]M. Born, Quantenmechanik der StoBvorgange[J], Z. f. Physik, vol.38,1926, pp.803-827.1
[17]W. Pauli, Zur Quantenmechanik des magnetischen Elektrons[J], Zeits. f. Phys., vol.43, 1927, pp.601-623 1
[18]P. A. M. Dirac, The Quantum Theory of the Electron[J], Proc. R. Soc. Lond. A., vol. 117,1928, pp.610-624.1
[19]H. Bethe, Quantum theory[J], Rev. Mod. Phys., vol.71,1999, pp:S1-S5.1,3
[20]P. A. M. Dirac, Quantum Mechanics of Many-Electron Systems[J], Proc. R. Soc. Lond. A, vol.123,1929, pp.714-733.1
[21]W. van Dijk and F. M. Toyama, Accurate numerical solutions of the time-dependent Schroinger equation[J], Phys. Rev. E, vol.75,2007,036707.3,109,118
[22]T.Iitaka, Solving the time-dependent Schrodinger equation numerically[J]. Phys. Rev. E, vol.49,1994, pp.4684-4690.109
[23]Ⅰ. Puzynin, A. Selin, and S. Vinitsky, A high-order accuracy method for numerical solv-ing of the time-dependent Schroinger equation[J], Comput. Phys. Commun., vol.123, 1999, pp.1-6.109,118
[24]Ⅰ. Puzynin, A. Selin, and S. Vinitsky, Magnus-factorized method for numerical solving the time-dependent Schrodinger equation[J], Comput. Phys. Commun., vol.126,2000, pp.158-161.109
[25](?). Mi(?)icu, M. Rizea, and W. Greiner, Emission of electromagnetic radiation in alpha decay[J], J. Phys. G, vol.27,2001, pp.993-1003.109
[26]W. S. Dias, E. M. Nascimento, M. L. Lyra, and F. A. B. F. de Moura, Frequency doubling of Bloch oscillations for interacting electrons in a static electric field[J], Phys. Rev. B, vol.76,2007,155124.3,109
[27]http://en.wikipedia.org/wiki/Pi 4
[28]Ch. Schwartz, Experiment and Theory in Computations of the He Atom Ground State[J], preprint at arXiv:physics/0208004,2002.4
[29]F. Bashforth and J. C. Adams, An attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of uid, with an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops[M], (Cambridge University Press, Cambridge,1883).4,6
[30]C. Runge, Uber die numerische Auflosung von Differentialgleichungen[J], Math. Ann., vol.46,1895,pp.167-178.4
[31]J. C. Butcher, Numerical methods for ordinary differential equations in the 20th cen-tury[J], J. Comp. Appl. Math., vol.125,2000, pp.1-29.4
[32]K. Heun, Neue Methoden zur approximativen Integration der Differentialgleichungen einer unabhangigen Veranderlichen[J], Z. Math. Phys. vol.45,1900, pp.23-38.4
[33]W. Kutta, Beitrag zur naherungsweisen Integration totaler Differentialgleichungen[J], Z. Math. Phys. vol.46,1901, pp.435-453.4
[34]E. J. Nystrom, Uber die numerische Integration von Differentialgleichungen[J], Acta Soc. Sci. Fenn., vol.50,1925, pp.1-54.4,6
[35]A. Huta, Une amelioration de la methode de Runge-Kutta-Nystrom pour la resolution numerique des equations differentielles du premier ordre[J], Acta Fac. Rerum Natur. Univ. Comenianae(Bratislava) Math., vol.1,1956, pp.201-224.4
[36]E. Hairer, S. P Nφrsett and G. Wanner, Solving Ordinary Differential Equations Ⅰ. Non-stiff Problems,2ed ed[M]. (Springer-Verlag Berlin Heidelberg.1987).4
[37]R. H. Merson, An operational method for the study of integration processes[R], Proc. Symp. Data Processing, Weapons Research Establishment, Salisbury, Australia,1957, pp.110.4
[38]F. Ceschino, Evaluation de l'erreur par pas dans les problemes differentiels[J], Chiffres, vol.5,1962, pp.223-229.4
[39]J. A. Zonneveld, Automatic integration of ordinary differential equations[R], Report R743, Mathematisch Centrum, Postbus 4079,1009AB Amsterdam,1963.4
[40]D. Sarafyan. Error estimation for Runge-Kutta methods through pseudo-iterative formu-las[R], Techn. Rep. No.14, Lousiana State Univ., New Orleans, May 1966.4
[41]R. England, Error estimates for Runge-Kutta type solutions to systems of ordinary dif-ferential equations[J], The Computer J.,vol.12,1969, pp.166-170.4
[42]E. Fehlberg, New high-order Runge-Kutta formulas with step size control for systems of first and second order differential equations[J], ZAMM, vol.44, Sonderheft T17-T19, 1964.4,5
[43]E. Fehlberg, Classical fifth-, sixth-, seventh-, and eighth order Runge-kutta formulas with step size control[R], NASA Technical Report 287,1968, extract published in Com-puting, vol.4,1969, pp.93-106.4
[44]E. Fehlberg, Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems[R], NASA Technical Report 315,1969, ex-tract published in Computing, vol.6,1970, pp.61-71.4
[45]J. C. Butcher, Coefficients for the study of Runge-Kutta integration processes[J], J. Aus-tral. Math. Soc. vol.3,1963, pp.185-201.5
[46]J. C. Butcher, On Runge-Kutta processes of high order[J], J. Austral. Math. Soc, vol. IV, Part2,1964, pp.179-194.5
[47]J. C. Butcher, On the attainable order of Runge-Kutta methods[J], Math. Comp. vol.19, 1965, pp.408-417.5
[48]J. C. Butcher, The non-existence of ten stage eighth order explicit Runge-Kutta meth-ods[J], BIT. vol.25,1985, pp.521-540.5
[49]A. R. Curtis, An eighth order Runge-Kutta process with eleven function evaluations per step[J], Numer. Math., vol.16,1970, pp.268-277.5
[50]G. J. Cooper and J. H. Verner, Some explicit Runge-Kutta methods of high order[J], SIAM J. Numer. Anal., vol.9,1972, pp.389-405.5
[51]A. R. Curtis, High-order explicit Runge-Kutta formulae, their uses, and limitations[J], J. Inst. Maths Applics, vol.16,1975, pp.35-55.5
[52]E. Hairer, A Runge-Kutta method of order 10[J], J. Inst. Maths Applics, vol.21,1978, pp.47-59.5
[53]A. L. Cauchy, Resume des le(?)ons donnees a l'Ecole Royale Polytechnique, Suite du Cal-cul Infinitesimal; published:Equations differentielles ordinaires, ed. Chr. Gilain, John-son,1981.5
[54]P. C. Hammer and J. W. Hollingsworth, Trapezoidal methods of approximating solutions of differential equations[J], MTAC, vol.9,1955, pp.92-96.5
[55]J. Kuntzmann, Neuere Entwickelungen der Methode von Runge-Kutta[J], ZAMM, vol. 41,1961, pp.28-31.5
[56]F. Ceschino and J. Kuntzmann, Numerical solutions of initial value problems[M], (Pren-tice Hall,1966). pp.106.5
[57]J. C. Butcher, Implicit Runge-Kutta Preocesses[J], Math. Comput., vol.18,1964, pp. 50-64.5
[58]J. C. Butcher, Integration processes based on Radau quadrature formulas[J], Math. Com-put., vol.18,1964, pp.233-244.5
[59]G. Darboux, Sur les developpements en serie des fonctions d'une seule variable, J. des Mathematique pures et appl.,3eme serie,1876, pp.291-312.5
[60]Ch. Hermite, Extrait d'une lettre de M. Ch. Hermite a M. Borchardt sur la formule d'interpolation de Lagrange[J], J. de Crelle, vol.84,1878, pp.432-443.5,19
[61]N. Obrechkoff, Neue Quadraturformeln, Abh. der Preuss. Akad. der Wiss., Math. na-truwiss. Klasse, Nr.4, Berlin.5,19
[62]F. R. Loscalzo and I. J. Schoenberg, On the use of spline functions for the approximation of solutions of ordinary differential equations[R], Tech. Summ. Rep.# 723, Math. Res. Center, Univ. Wisconsin, Madison.1967.5
[63]S. P. Nφsett, One-step methods of Hermite type for numerical integration of stiff sys-tems[J], BIT, vol.14,1974, pp.63-77.5
[64]R. Zurmiihl, Runge-Kutta Verfahren unter Verwendung hoherer Ableitungen[J], ZAMM, vol.32,1952, pp.153-154.5
[65]J. Albrecht, Beitrage zum Runge-Kutta-Verfahren[J], ZAMM, vol.35,1955, pp.100-110.5
[66]E. Fehlberg, Eine Methode zur Fehlerverkleinerung beim Runge-Kutta-Verfahren[J], ZAMM, vol.38,1958, pp.421-426.5
[67]K. H. Kastlunger and G. Wanner, Runge Kutta processes with multiple nodes[J], Com-puting, vol.9,1972, pp.9-24.5
[68]K. H. Kastlunger and G.Wanner, On Turan type implicit Runge-Kutta methods[J], Com-puting, vol.9,1972, pp.317-325.5
[69]E. Hairer and G. Wanner, On the Butcher group and general multi-value methods[J], Computing, vol.13,1974, pp.1-15.5
[70]F. R. Moulton, New methods in exterior ballistics[M], (Univ. Chicago Press,1926).6
[71]W. E. Milne, Numerical integration of ordinary differential equations[J], Amer. Math. Monthly, vol.33,1926, pp.455-460.6
[72]C. F. Curtiss and J. O. Hirschfelder, Integration of stiff equations[J], Proc. of the National Academy of Sciences of U.S., vol.38,1952, pp.235-243.6
[73]C. W. Gear, Numerical initial value problems in ordinary differential equations[M], (Prentice-Hall,1971).6
[74]G. Dahlquist, Convergence and stability in the numerical integration of ordinary differ-ential equations[J], Math. Scand., vol.4,1956, pp.33-53.6
[75]G. Dahlquist, Stability and error bounds in the numerical integration of ordinary differ-ential equations[C], Trans. of the Royal Inst. of Techn., Stockholm, Sweden, Nr.130, 1959.6
[76]P. Henrici, Discrete variable methods in ordinary differential equations[M], (John Wiley & Sons, Inc., New-York-London-Sydney.1962).6
[77]H. Rutishauser, Ueber die Instabilitat von Methoden zur Integration gewohnlicher Dif-ferentialgleichungen[J]. ZAMP, vol.3,1952, pp.65-74.6
[78]J. Todd, Notes on modern numerical analysis, I[J], Math. Tables Aids Comput., vol.4, 1950, pp.39-44.6
[79]A. R. Mitchell and J. W. Craggs, Stability of difference relations in the solution of or-dinary differential equations[J], Math. Tables Aids Comput., vol.7,1953, pp.127-129. 6
[80]C. W. Cryer, A proof of the instability of backward-difference multistep methods for the numerical integration of ordinary differential equations[R], Tech. Rep. No.117, Comp. Sci. Dept., Univ. of Wisconsin,1971, pp.1-52.6
[81]D. M. Creedon and J. J. H. Miller, The stability properties of q-step backward-difference schemes[J]. BIT, vol.15,1975, pp.244-249.6
[82]E. Hairer and G. Wanner, On the instability of the BDF formulas[J], SIAM J. Numer. Anal., vol.20, No.6,1983, pp.1206-1209.6
[83]R. von Mises, Zur numerischen Integration von Differentialgleichungen[J], ZAMM, vol. 10,1930, pp.81-92.6
[84]W. Tollmien, Ueber die Fehlerabschatzung beim Adamsschen Verfahren zur Integration gewohnlicher Differentialgleichungen[J], ZAMM, vol.18,1938, pp.83-90.6
[85]A. Fricke, Ueber die Fehlerabschatzung des Adamsschen Verfahrens zur Integration gewohnlicher Differentialgleichungen erster Ordnung[J], ZAMM, vol.29,1949, pp. 165-178.6
[86]J. Weissinger, Eine verscharfte Fehlerabschatzung zum Extrapolationsverfahren von Adams[J], ZAMM, vol.30,1950, pp.356-363.6
[87]L. Vietoris, Der Richtungsfehler einer durch das Adamssche Interpolationsverfahren gewonnenen Naherungslosung einer Gleichung y"=f(x,y), Oesterr[J]. Akad. Wiss., Math.-naturw. Kl., Abt. Ⅱa, vol.162,1953, pp.157-167 and pp.293-299.6
[88]J. C. Butcher, On the convergence of numerical solutions to ordinary differential equa-tions[J], Math. Comput., vol.20,1966, pp.1-10.6
[89]F. Ceschino, Modification de la longueur du pas dans 1'integration numerique par les methodes a pas lies[J], Chiffres, vol.2,1961, pp.101-106.6
[90]C. V. D. Forrington, Extensions of the predictor-corrector method for the solution of systems of ordinary differential equations[J], Comput. J. vol.4,1961, pp.80-84.6
[91]F. T. Krogh, A variable step variable order multistep method for the numerical solution of ordinary differential equations [A], Information Processing 68[C], North-Holland, Ams-terdam,1969, pp.194-199.6
[92]C. W. Gear and K. W. Tu, The effect of variable mesh size on the stability of multistep methods[J], SIAM J. Num. Anal., vol.11,1974, pp.1025-1043.6
[93]C. W. Gear and D. S. Watanabe, Stability and convergence of variable order multistep methods[J]. SIAM J. Num. Anal., vol.11,1974, pp.1044-1058.6
[94]R. D. Grigorieff, Stability of multistep-methods on variable grids [J], Numer. Math. vol. 42,1983, pp.359-377.7
[95]M. Crouzeix and F.J. Lisbona, The convergence of variable-stepsize, variable formula, multistep methods[J], SIAM J. Num. Anal., vol.21,1984, pp.512-534.7
[96]P. Piotrowsky, Stability, consistency and convergence of variable k -step methods for nu-merical integration of large systems of ordinary differential equations [A], Lecture Notes in Math.[M],109, Dundee 1969, pp.221-227.7
[97]A. Nordsieck, On numerical integration of ordinary differential equations[J]. Math. Comp., vol.16,1962, pp.22-49.7
[98]J. Descloux, A note on a paper by A. Nordsieck[R]. Report No.131, Dept. of Comp. Sci., Univ. of Illinois at Urbana-Champaign.1963.7
[99]M. R. Osborne, On Nordsieck's method for the numerical solution of ordinary differ-ential equations [J]. BIT, vol.6,1966, pp.51-57.7
[100]R. D. Skeel, Equivalent forms of multistep formulas[J]. Math. Comput., vol.33,1979, pp.1229-1250.7
[101]W. B. Gragg, Repeated extrapolation to the limit in the numerical solution of ordinary differential equations[D], Thesis, (Univ. of California,1964).7
[102]J. C. Butcher, A modified multistep method for the numerical integration of ordinary differential equations [J], J. ACM, vol.12,1965, pp.124-135.7
[103]C. W. Gear, Hybrid methods for initial value problems in ordinary differential equa-tions[J], SIAM J. Numer. Anal., ser. B, vol.2,1965, pp.69-86.7
[104]G. D. Byrne and R. J. Lambert, Pseudo-Runge-Kutta methods involving two points[J]. J. Assoc. Comput. Mach., vol.13,1966, pp.114-123.7
[105]J. Donelson and E. Hansen, Cyclic composite multistep predictor-corrector methods[J]. SIAM, J. Numer. Anal., vol.8,1971, pp.137-157.7
[106]K. Burrage and J.C. Butcher, Non-linear stability of a general class of differential equa-tion methods[J], BIT, vol.20,1980, pp.185-203.7
[107]C. Stormer, Sur les trajectoires des corpuscules electrises, Arch. sci. phys. nat., Geneve, vol.24,1907, pp.5-18,113-158,221-247.7
[108]P. H. Cowell and A. C. D. Crommelin, Investigation of the motion of Halley's comet from 1759 to 1910[M], Appendix to Greenwich Observations for 1909, Edinburgh, 1910, pp.1-84.7
[109]B. Numerov, A method of extrapolation of perturbations [J]. Monthly notices of the Royal Astronomical Society, vol.84,1924, pp.592-601.7,93
[110]G. E. Kimball and G. H. Shortley, The numerical solutions of Schroinger equation[J], Phys. Rev., vol.45,1934, pp.815-820.7
[111]L. Gr. Ixaru, G. Vanden Berghe, Exponential Fitting[M], (Kluwer Academic Publish-ers,2004).8
[112]G. Vanden Berghe, H. De Meyer, M. Van Daele, T. Van Hecke, Exponentially fitted Runge-Kutta methods[J], J. Comput. Appl. Math., vol.125,2000, pp.107-115.8
[113]T. E. Simos, Atomic Structure Computations in Chemical Modelling:Applications and Theory, Vol 1 (Editor:A. Hinchliffe, UMIST)[M], The Royal Society of Chemistry, 2000, pp.38-142.8
[114]L. Gr. Ixaru, CP methods for the Schrodinger equation[J], J. Comput. Appl. Math., vol. 125,2000, pp.347-357.8
[115]A.J. Zakrzewski, Highly precise solutions of the one-dimensional Schroinger equation with an arbitrary potential[J], Comput. Phys. Commun., vol.175,2006, pp.397-403.8, 21
[1]Ch. Hermite, Extrait d'une lettre de M. Ch. Hermite a M. Borchardt sur la formule d'interpolation de Lagrange[J], J. de Crelle, vol.84,1878, pp.432-443.5,19
[2]N. Obrechkoff, Neue Quadraturformeln, Abh. der Preuss. Akad. der Wiss., Math. na-truwiss. Klasse, Nr.4, Berlin.5,19
[3]G. Dahlquist,33 Years of Numerical Instability, Part Ⅰ[J], BIT, vol.25,1985, pp.188-204.19
[4]R. Courant, K. O. Friedrichs and H. Lewy, Uber die partiellen Differenzengleichungen der mathematischen Physik[J], Math. Ann. vol.100,1928, pp.32-74.19
[5]L. Collatz and R. Zurmuhl,Beitrage zu den Interpolationsverfahren der numerischen In-tegration von Differentialgleichungen erster und zweiter Ordnung[J], ZAMM, vol.22, 1942, pp.42-55.20
[6]J. von Neumann and H. H. Goldstine, Numerical inverting of matrices of high order[J], BAMS, vol.53,1947, pp.1021-1099.20
[7]J. Crank and P. Nicolson, A practical method for numerical integration of solutions of partial differential equations of heat-conduction type[J], Proc. Cambridge Phil. Soc., vol. 43,1947, pp.50-67.20
[8]J. Crank and P. Nicolson, A practical method for numerical integration of solutions of partial differential equations of heat-conduction type[J], Advances in Computational Mathematics, vol.6,1996, pp.207-226.20,109
[9]J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial value problems[J], J. Inst. Math. Applic., vol.18,1976, pp.189-202.20,55
[10]M. Van Daele and G. Vanden Berghe, P-stable Obrechkoff methods of arbitrary order for second-order differential equations[J], Numer. Algor., vol.44,2007, pp.115-131.20, 55,56
[11]Z. Wang, Y. Ge, Y. Dai, D. Zhao, A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.160,2004, pp.23-45.20,93,95
[12]D. Zhao, Z. Wang, Y. Dai, Importance of the first-order derivative formula in the Obrechkoff method[J], Comput. Phys. Commun., vol.167,2005, pp.65-75.21
[13]Z. Wang and Q. Chen, A trigonometrically-fitted one-step method with multi-derivative for the numerical solution to the one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.170,2005, pp.49-64.21,45
[14]A.J. Zakrzewski, Highly precise solutions of the one-dimensional Schroinger equation with an arbitrary potential[J], Comput. Phys. Commun., vol.175,2006, pp.397-403.8, 21
[15]冯康等编,数值计算方法[M],(国防工业出版社,1978),pp.473.24
[16]H. Shao and Z. Wang, Arbitrarily precise numerical solutions of the one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.180,2009, pp.1-7.37,38,41, 43,50,61,103
[17]J. Chen, Z. Wang, H. Shao, and H. Hao, Highly-accurate ground state energies of the He atom and the He-like ions by Hartree SCF calculation with Obrechkoff method[J], Comput. Phys. Commun., vol.179,2008, pp.486-491.38
[1]J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial value problems[J], J. Inst. Math. Applic., vol.18,1976, pp.189-202.20,55
[2]M. Van Daele and G. Vanden Berghe, P-stable Obrechkoff methods of arbitrary order for second-order differential equations[J], Numer. Algor., vol.44,2007, pp.115-131.20, 55,56
[3]U. Ananthakrishnaiah, P-stable Obrechkoff methods with minimal phase-lag for periodic initial value problems[J], Math. Comput. vol.49,1987, pp.553-559.55
[4]L. Brusa. and L. Nigro, A one-step method for direct integration of structural dynamic equations[J], Int. J. Numer. Meth. Engng., vol.15,1980, pp.553-559.56,102
[5]H. Shao and Z. Wang, Arbitrarily precise numerical solutions of the one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.180,2009, pp.1-7.37,38,41, 43,50,61,103
[6]P. M. Morse, Diatomic molecules according to the wave mechanics. Ⅱ. Vibrational lev-els[J], Phys. Rev., vol.34,1929, pp.57-64.63
[7]J. D. Praeger, Relaxational approach to solving the Schrodinger equation[J], Phys. Rev. A., vol.63,2001,022115.64
[8]S. Fliigge, Practical Quantum Mechanics[M]. (Berlin:Springer-Verlag,1974), pp.94. 65,105
[9]王竹溪,郭敦仁著,特殊函数概论[M],(北京大学出版社,2000),pp.131,137,288,289.64,66
[10]T. E. Stiefel and D. G. Bettis, Stabilization of Cowell's method[J], Numer. Math. vol. 13,1969, pp.154-175.68
[11]R. van Dooren, Stabilization of Cowell's classic finite difference method for numerical integration[J], J. Comput. Phys., vol.16,1974, pp.186-192.69
[12]Y. Dai, Z. Wang, D. Zhao, and X. Song, A new high efficient and high accurate Obrechkoff four-step method for the periodic nonlinear undamped duffing's equation[J], Comput. Phys. Commun., vol.165,2005, pp.110-126.70
[13]周纪卿,朱因远,非线性振动[M],(西安交通大学出版社,1998),pp.186-187.71
[14]D. Wu and Z. Wang, A Mathematica Program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach[J]. Computer Physics Communications, vol.174,2005, pp.447-463.75
[1]B. Numerov, A method of extrapolation of perturbations[J]. Monthly notices of the Royal Astronomical Society, vol.84,1924, pp.592-601.7,93
[2]Z. Wang, Y. Ge, Y. Dai, D. Zhao, A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.160,2004, pp.23-45.20,93,95
[3]L. Brusa. and L. Nigro, A one-step method for direct integration of structural dynamic equations[J], Int. J. Numer. Meth. Engng., vol.15,1980, pp.553-559.56,102
[4]H. Shao and Z. Wang, Arbitrarily precise numerical solutions of the one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.180,2009, pp.1-7.37,38,41, 43,50,61,103
[5]S. Fliigge, Practical Quantum Mechanics[M]. (Berlin:Springer-Verlag,1974), pp.94. 65,105
[1]J. Crank and P. Nicolson, A practical method for numerical integration of solutions of partial differential equations of heat-conduction type[J], Advances in Computational Mathematics, vol.6,1996, pp.207-226.20,109
[2]W. van Dijk and F. M. Toyama, Accurate numerical solutions of the time-dependent Schroinger equation[J], Phys. Rev. E, vol.75,2007,036707.3,109,118
[3]T.Iitaka, Solving the time-dependent Schrodinger equation numerically[J]. Phys. Rev. E, vol.49,1994, pp.4684-4690.109
[4]I. Puzynin, A. Selin, and S. Vinitsky, A high-order accuracy method for numerical solv-ing of the time-dependent Schroinger equation[J], Comput. Phys. Commun., vol.123, 1999, pp.1-6.109,118
[5]I. Puzynin, A. Selin, and S. Vinitsky, Magnus-factorized method for numerical solving the time-dependent Schrodinger equation[J], Comput. Phys. Commun., vol.126,2000, pp.158-161.109
[6](?). Mi(?)icu, M. Rizea, and W. Greiner, Emission of electromagnetic radiation in alpha decay[J], J. Phys. G, vol.27,2001, pp.993-1003.109
[7]W. S. Dias, E. M. Nascimento, M. L. Lyra, and F. A. B. F. de Moura, Frequency doubling of Bloch oscillations for interacting electrons in a static electric field[J], Phys. Rev. B, vol.76,2007,155124.3,109
[8]S. Wolfram, The Mathematica Book,5th ed[M]. (Wolfram Media, Inc.2003).
[9]A. Goldberg, H. M. Schey, and J. L. Schwartz, Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena[J], Am. J. Phys., vol.35,1967, pp.177-186.118
[10]L. I. Schiff, Quantum Mechanics, International Series in Pure and Applied Physics,1st ed[M]. (McGraw-Hill Book Company Inc., New York,1949), pp:67.118,119,121
[11]T. Cheon, I. Tsutsui, and T. Fulop, Quantum Abacus[J], Phys. Lett. A, vol.330,2004, pp:338-342.119
[2]J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Vol.1: The Quantum Theory of Planck, Einstein, Bohr and Sommerfeld. Its Foundation and the Rise of Its Difficulties (1900-1925)[M]. (New York:Springer-Verlag,1982).1
[3]M. Planck, On the Law of Distribution of Energy in the Normal Spectrum[J], Ann. Physik, vol.4,1901, pp.553 ff.1
[4]A. Einstein, On a Heuristic Viewpoint Concerning the Production and Transformation of Light[J], Ann. Physik, vol.17,1905, pp.132-148.1
[5]曾谨言著,量子力学卷二[M],(科学出版社,1997),pp.1-2.1
[6]W. Heisenberg, Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen[J], Z. f. Physik, vol.33,1925, pp.879-893.1
[7]M. Born and P. Jordan, Zur Quantenmechanik[J], Z. f. Physik, vol.34,1925, pp.858-888.
[8]M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik Ⅱ[J], Z. f. Physik, vol. 35,1925, pp.557-615.1
[9]L. de Broglie, Recherches sur la theorie des quanta (Researches on the quantum the-ory)[D], Thesis, (Paris,1924).1
[10]E. Schrodinger, An Undulatory Theory of the Mechanics of Atoms and Molecules[J], Phys. Rev., vol.28,1926, pp.1049-1070.1
[11]J. Mehra, Erwin Schrodinger and the Rise of Wave Mechanics. Ⅰ. Schrodinger's Scien-tific Work before the Creation of Wave Mechanics[J], Foundations of Physics, vol.17, 1987, pp.1051-1112.
[12]J. Mehra, Erwin Schrodinger and the Rise of Wave Mechanics. Ⅱ. The Creation of Wave Mechanics[J], Foundations of Physics, vol.17,1987, pp.1141-1188.
[13]J. Mehra, Erwin Schrodinger and the Rise of Wave Mechanics. Ⅲ. Early Response and Applications[J], Foundations of Physics, vol.18,1988, pp.107-184.1
[14]P. A. M. Dirac, Ph.D. Thesis[D]. (Cambridge University,1926).1
[15]M. Born, Zur Quantenmechanik der Stoβvorgange[J], Z. f. Physik, vol.37,1926, pp. 863-867.1
[16]M. Born, Quantenmechanik der StoBvorgange[J], Z. f. Physik, vol.38,1926, pp.803-827.1
[17]W. Pauli, Zur Quantenmechanik des magnetischen Elektrons[J], Zeits. f. Phys., vol.43, 1927, pp.601-623 1
[18]P. A. M. Dirac, The Quantum Theory of the Electron[J], Proc. R. Soc. Lond. A., vol. 117,1928, pp.610-624.1
[19]H. Bethe, Quantum theory[J], Rev. Mod. Phys., vol.71,1999, pp:S1-S5.1,3
[20]P. A. M. Dirac, Quantum Mechanics of Many-Electron Systems[J], Proc. R. Soc. Lond. A, vol.123,1929, pp.714-733.1
[21]W. van Dijk and F. M. Toyama, Accurate numerical solutions of the time-dependent Schroinger equation[J], Phys. Rev. E, vol.75,2007,036707.3,109,118
[22]T.Iitaka, Solving the time-dependent Schrodinger equation numerically[J]. Phys. Rev. E, vol.49,1994, pp.4684-4690.109
[23]Ⅰ. Puzynin, A. Selin, and S. Vinitsky, A high-order accuracy method for numerical solv-ing of the time-dependent Schroinger equation[J], Comput. Phys. Commun., vol.123, 1999, pp.1-6.109,118
[24]Ⅰ. Puzynin, A. Selin, and S. Vinitsky, Magnus-factorized method for numerical solving the time-dependent Schrodinger equation[J], Comput. Phys. Commun., vol.126,2000, pp.158-161.109
[25](?). Mi(?)icu, M. Rizea, and W. Greiner, Emission of electromagnetic radiation in alpha decay[J], J. Phys. G, vol.27,2001, pp.993-1003.109
[26]W. S. Dias, E. M. Nascimento, M. L. Lyra, and F. A. B. F. de Moura, Frequency doubling of Bloch oscillations for interacting electrons in a static electric field[J], Phys. Rev. B, vol.76,2007,155124.3,109
[27]http://en.wikipedia.org/wiki/Pi 4
[28]Ch. Schwartz, Experiment and Theory in Computations of the He Atom Ground State[J], preprint at arXiv:physics/0208004,2002.4
[29]F. Bashforth and J. C. Adams, An attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of uid, with an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops[M], (Cambridge University Press, Cambridge,1883).4,6
[30]C. Runge, Uber die numerische Auflosung von Differentialgleichungen[J], Math. Ann., vol.46,1895,pp.167-178.4
[31]J. C. Butcher, Numerical methods for ordinary differential equations in the 20th cen-tury[J], J. Comp. Appl. Math., vol.125,2000, pp.1-29.4
[32]K. Heun, Neue Methoden zur approximativen Integration der Differentialgleichungen einer unabhangigen Veranderlichen[J], Z. Math. Phys. vol.45,1900, pp.23-38.4
[33]W. Kutta, Beitrag zur naherungsweisen Integration totaler Differentialgleichungen[J], Z. Math. Phys. vol.46,1901, pp.435-453.4
[34]E. J. Nystrom, Uber die numerische Integration von Differentialgleichungen[J], Acta Soc. Sci. Fenn., vol.50,1925, pp.1-54.4,6
[35]A. Huta, Une amelioration de la methode de Runge-Kutta-Nystrom pour la resolution numerique des equations differentielles du premier ordre[J], Acta Fac. Rerum Natur. Univ. Comenianae(Bratislava) Math., vol.1,1956, pp.201-224.4
[36]E. Hairer, S. P Nφrsett and G. Wanner, Solving Ordinary Differential Equations Ⅰ. Non-stiff Problems,2ed ed[M]. (Springer-Verlag Berlin Heidelberg.1987).4
[37]R. H. Merson, An operational method for the study of integration processes[R], Proc. Symp. Data Processing, Weapons Research Establishment, Salisbury, Australia,1957, pp.110.4
[38]F. Ceschino, Evaluation de l'erreur par pas dans les problemes differentiels[J], Chiffres, vol.5,1962, pp.223-229.4
[39]J. A. Zonneveld, Automatic integration of ordinary differential equations[R], Report R743, Mathematisch Centrum, Postbus 4079,1009AB Amsterdam,1963.4
[40]D. Sarafyan. Error estimation for Runge-Kutta methods through pseudo-iterative formu-las[R], Techn. Rep. No.14, Lousiana State Univ., New Orleans, May 1966.4
[41]R. England, Error estimates for Runge-Kutta type solutions to systems of ordinary dif-ferential equations[J], The Computer J.,vol.12,1969, pp.166-170.4
[42]E. Fehlberg, New high-order Runge-Kutta formulas with step size control for systems of first and second order differential equations[J], ZAMM, vol.44, Sonderheft T17-T19, 1964.4,5
[43]E. Fehlberg, Classical fifth-, sixth-, seventh-, and eighth order Runge-kutta formulas with step size control[R], NASA Technical Report 287,1968, extract published in Com-puting, vol.4,1969, pp.93-106.4
[44]E. Fehlberg, Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems[R], NASA Technical Report 315,1969, ex-tract published in Computing, vol.6,1970, pp.61-71.4
[45]J. C. Butcher, Coefficients for the study of Runge-Kutta integration processes[J], J. Aus-tral. Math. Soc. vol.3,1963, pp.185-201.5
[46]J. C. Butcher, On Runge-Kutta processes of high order[J], J. Austral. Math. Soc, vol. IV, Part2,1964, pp.179-194.5
[47]J. C. Butcher, On the attainable order of Runge-Kutta methods[J], Math. Comp. vol.19, 1965, pp.408-417.5
[48]J. C. Butcher, The non-existence of ten stage eighth order explicit Runge-Kutta meth-ods[J], BIT. vol.25,1985, pp.521-540.5
[49]A. R. Curtis, An eighth order Runge-Kutta process with eleven function evaluations per step[J], Numer. Math., vol.16,1970, pp.268-277.5
[50]G. J. Cooper and J. H. Verner, Some explicit Runge-Kutta methods of high order[J], SIAM J. Numer. Anal., vol.9,1972, pp.389-405.5
[51]A. R. Curtis, High-order explicit Runge-Kutta formulae, their uses, and limitations[J], J. Inst. Maths Applics, vol.16,1975, pp.35-55.5
[52]E. Hairer, A Runge-Kutta method of order 10[J], J. Inst. Maths Applics, vol.21,1978, pp.47-59.5
[53]A. L. Cauchy, Resume des le(?)ons donnees a l'Ecole Royale Polytechnique, Suite du Cal-cul Infinitesimal; published:Equations differentielles ordinaires, ed. Chr. Gilain, John-son,1981.5
[54]P. C. Hammer and J. W. Hollingsworth, Trapezoidal methods of approximating solutions of differential equations[J], MTAC, vol.9,1955, pp.92-96.5
[55]J. Kuntzmann, Neuere Entwickelungen der Methode von Runge-Kutta[J], ZAMM, vol. 41,1961, pp.28-31.5
[56]F. Ceschino and J. Kuntzmann, Numerical solutions of initial value problems[M], (Pren-tice Hall,1966). pp.106.5
[57]J. C. Butcher, Implicit Runge-Kutta Preocesses[J], Math. Comput., vol.18,1964, pp. 50-64.5
[58]J. C. Butcher, Integration processes based on Radau quadrature formulas[J], Math. Com-put., vol.18,1964, pp.233-244.5
[59]G. Darboux, Sur les developpements en serie des fonctions d'une seule variable, J. des Mathematique pures et appl.,3eme serie,1876, pp.291-312.5
[60]Ch. Hermite, Extrait d'une lettre de M. Ch. Hermite a M. Borchardt sur la formule d'interpolation de Lagrange[J], J. de Crelle, vol.84,1878, pp.432-443.5,19
[61]N. Obrechkoff, Neue Quadraturformeln, Abh. der Preuss. Akad. der Wiss., Math. na-truwiss. Klasse, Nr.4, Berlin.5,19
[62]F. R. Loscalzo and I. J. Schoenberg, On the use of spline functions for the approximation of solutions of ordinary differential equations[R], Tech. Summ. Rep.# 723, Math. Res. Center, Univ. Wisconsin, Madison.1967.5
[63]S. P. Nφsett, One-step methods of Hermite type for numerical integration of stiff sys-tems[J], BIT, vol.14,1974, pp.63-77.5
[64]R. Zurmiihl, Runge-Kutta Verfahren unter Verwendung hoherer Ableitungen[J], ZAMM, vol.32,1952, pp.153-154.5
[65]J. Albrecht, Beitrage zum Runge-Kutta-Verfahren[J], ZAMM, vol.35,1955, pp.100-110.5
[66]E. Fehlberg, Eine Methode zur Fehlerverkleinerung beim Runge-Kutta-Verfahren[J], ZAMM, vol.38,1958, pp.421-426.5
[67]K. H. Kastlunger and G. Wanner, Runge Kutta processes with multiple nodes[J], Com-puting, vol.9,1972, pp.9-24.5
[68]K. H. Kastlunger and G.Wanner, On Turan type implicit Runge-Kutta methods[J], Com-puting, vol.9,1972, pp.317-325.5
[69]E. Hairer and G. Wanner, On the Butcher group and general multi-value methods[J], Computing, vol.13,1974, pp.1-15.5
[70]F. R. Moulton, New methods in exterior ballistics[M], (Univ. Chicago Press,1926).6
[71]W. E. Milne, Numerical integration of ordinary differential equations[J], Amer. Math. Monthly, vol.33,1926, pp.455-460.6
[72]C. F. Curtiss and J. O. Hirschfelder, Integration of stiff equations[J], Proc. of the National Academy of Sciences of U.S., vol.38,1952, pp.235-243.6
[73]C. W. Gear, Numerical initial value problems in ordinary differential equations[M], (Prentice-Hall,1971).6
[74]G. Dahlquist, Convergence and stability in the numerical integration of ordinary differ-ential equations[J], Math. Scand., vol.4,1956, pp.33-53.6
[75]G. Dahlquist, Stability and error bounds in the numerical integration of ordinary differ-ential equations[C], Trans. of the Royal Inst. of Techn., Stockholm, Sweden, Nr.130, 1959.6
[76]P. Henrici, Discrete variable methods in ordinary differential equations[M], (John Wiley & Sons, Inc., New-York-London-Sydney.1962).6
[77]H. Rutishauser, Ueber die Instabilitat von Methoden zur Integration gewohnlicher Dif-ferentialgleichungen[J]. ZAMP, vol.3,1952, pp.65-74.6
[78]J. Todd, Notes on modern numerical analysis, I[J], Math. Tables Aids Comput., vol.4, 1950, pp.39-44.6
[79]A. R. Mitchell and J. W. Craggs, Stability of difference relations in the solution of or-dinary differential equations[J], Math. Tables Aids Comput., vol.7,1953, pp.127-129. 6
[80]C. W. Cryer, A proof of the instability of backward-difference multistep methods for the numerical integration of ordinary differential equations[R], Tech. Rep. No.117, Comp. Sci. Dept., Univ. of Wisconsin,1971, pp.1-52.6
[81]D. M. Creedon and J. J. H. Miller, The stability properties of q-step backward-difference schemes[J]. BIT, vol.15,1975, pp.244-249.6
[82]E. Hairer and G. Wanner, On the instability of the BDF formulas[J], SIAM J. Numer. Anal., vol.20, No.6,1983, pp.1206-1209.6
[83]R. von Mises, Zur numerischen Integration von Differentialgleichungen[J], ZAMM, vol. 10,1930, pp.81-92.6
[84]W. Tollmien, Ueber die Fehlerabschatzung beim Adamsschen Verfahren zur Integration gewohnlicher Differentialgleichungen[J], ZAMM, vol.18,1938, pp.83-90.6
[85]A. Fricke, Ueber die Fehlerabschatzung des Adamsschen Verfahrens zur Integration gewohnlicher Differentialgleichungen erster Ordnung[J], ZAMM, vol.29,1949, pp. 165-178.6
[86]J. Weissinger, Eine verscharfte Fehlerabschatzung zum Extrapolationsverfahren von Adams[J], ZAMM, vol.30,1950, pp.356-363.6
[87]L. Vietoris, Der Richtungsfehler einer durch das Adamssche Interpolationsverfahren gewonnenen Naherungslosung einer Gleichung y"=f(x,y), Oesterr[J]. Akad. Wiss., Math.-naturw. Kl., Abt. Ⅱa, vol.162,1953, pp.157-167 and pp.293-299.6
[88]J. C. Butcher, On the convergence of numerical solutions to ordinary differential equa-tions[J], Math. Comput., vol.20,1966, pp.1-10.6
[89]F. Ceschino, Modification de la longueur du pas dans 1'integration numerique par les methodes a pas lies[J], Chiffres, vol.2,1961, pp.101-106.6
[90]C. V. D. Forrington, Extensions of the predictor-corrector method for the solution of systems of ordinary differential equations[J], Comput. J. vol.4,1961, pp.80-84.6
[91]F. T. Krogh, A variable step variable order multistep method for the numerical solution of ordinary differential equations [A], Information Processing 68[C], North-Holland, Ams-terdam,1969, pp.194-199.6
[92]C. W. Gear and K. W. Tu, The effect of variable mesh size on the stability of multistep methods[J], SIAM J. Num. Anal., vol.11,1974, pp.1025-1043.6
[93]C. W. Gear and D. S. Watanabe, Stability and convergence of variable order multistep methods[J]. SIAM J. Num. Anal., vol.11,1974, pp.1044-1058.6
[94]R. D. Grigorieff, Stability of multistep-methods on variable grids [J], Numer. Math. vol. 42,1983, pp.359-377.7
[95]M. Crouzeix and F.J. Lisbona, The convergence of variable-stepsize, variable formula, multistep methods[J], SIAM J. Num. Anal., vol.21,1984, pp.512-534.7
[96]P. Piotrowsky, Stability, consistency and convergence of variable k -step methods for nu-merical integration of large systems of ordinary differential equations [A], Lecture Notes in Math.[M],109, Dundee 1969, pp.221-227.7
[97]A. Nordsieck, On numerical integration of ordinary differential equations[J]. Math. Comp., vol.16,1962, pp.22-49.7
[98]J. Descloux, A note on a paper by A. Nordsieck[R]. Report No.131, Dept. of Comp. Sci., Univ. of Illinois at Urbana-Champaign.1963.7
[99]M. R. Osborne, On Nordsieck's method for the numerical solution of ordinary differ-ential equations [J]. BIT, vol.6,1966, pp.51-57.7
[100]R. D. Skeel, Equivalent forms of multistep formulas[J]. Math. Comput., vol.33,1979, pp.1229-1250.7
[101]W. B. Gragg, Repeated extrapolation to the limit in the numerical solution of ordinary differential equations[D], Thesis, (Univ. of California,1964).7
[102]J. C. Butcher, A modified multistep method for the numerical integration of ordinary differential equations [J], J. ACM, vol.12,1965, pp.124-135.7
[103]C. W. Gear, Hybrid methods for initial value problems in ordinary differential equa-tions[J], SIAM J. Numer. Anal., ser. B, vol.2,1965, pp.69-86.7
[104]G. D. Byrne and R. J. Lambert, Pseudo-Runge-Kutta methods involving two points[J]. J. Assoc. Comput. Mach., vol.13,1966, pp.114-123.7
[105]J. Donelson and E. Hansen, Cyclic composite multistep predictor-corrector methods[J]. SIAM, J. Numer. Anal., vol.8,1971, pp.137-157.7
[106]K. Burrage and J.C. Butcher, Non-linear stability of a general class of differential equa-tion methods[J], BIT, vol.20,1980, pp.185-203.7
[107]C. Stormer, Sur les trajectoires des corpuscules electrises, Arch. sci. phys. nat., Geneve, vol.24,1907, pp.5-18,113-158,221-247.7
[108]P. H. Cowell and A. C. D. Crommelin, Investigation of the motion of Halley's comet from 1759 to 1910[M], Appendix to Greenwich Observations for 1909, Edinburgh, 1910, pp.1-84.7
[109]B. Numerov, A method of extrapolation of perturbations [J]. Monthly notices of the Royal Astronomical Society, vol.84,1924, pp.592-601.7,93
[110]G. E. Kimball and G. H. Shortley, The numerical solutions of Schroinger equation[J], Phys. Rev., vol.45,1934, pp.815-820.7
[111]L. Gr. Ixaru, G. Vanden Berghe, Exponential Fitting[M], (Kluwer Academic Publish-ers,2004).8
[112]G. Vanden Berghe, H. De Meyer, M. Van Daele, T. Van Hecke, Exponentially fitted Runge-Kutta methods[J], J. Comput. Appl. Math., vol.125,2000, pp.107-115.8
[113]T. E. Simos, Atomic Structure Computations in Chemical Modelling:Applications and Theory, Vol 1 (Editor:A. Hinchliffe, UMIST)[M], The Royal Society of Chemistry, 2000, pp.38-142.8
[114]L. Gr. Ixaru, CP methods for the Schrodinger equation[J], J. Comput. Appl. Math., vol. 125,2000, pp.347-357.8
[115]A.J. Zakrzewski, Highly precise solutions of the one-dimensional Schroinger equation with an arbitrary potential[J], Comput. Phys. Commun., vol.175,2006, pp.397-403.8, 21
[1]Ch. Hermite, Extrait d'une lettre de M. Ch. Hermite a M. Borchardt sur la formule d'interpolation de Lagrange[J], J. de Crelle, vol.84,1878, pp.432-443.5,19
[2]N. Obrechkoff, Neue Quadraturformeln, Abh. der Preuss. Akad. der Wiss., Math. na-truwiss. Klasse, Nr.4, Berlin.5,19
[3]G. Dahlquist,33 Years of Numerical Instability, Part Ⅰ[J], BIT, vol.25,1985, pp.188-204.19
[4]R. Courant, K. O. Friedrichs and H. Lewy, Uber die partiellen Differenzengleichungen der mathematischen Physik[J], Math. Ann. vol.100,1928, pp.32-74.19
[5]L. Collatz and R. Zurmuhl,Beitrage zu den Interpolationsverfahren der numerischen In-tegration von Differentialgleichungen erster und zweiter Ordnung[J], ZAMM, vol.22, 1942, pp.42-55.20
[6]J. von Neumann and H. H. Goldstine, Numerical inverting of matrices of high order[J], BAMS, vol.53,1947, pp.1021-1099.20
[7]J. Crank and P. Nicolson, A practical method for numerical integration of solutions of partial differential equations of heat-conduction type[J], Proc. Cambridge Phil. Soc., vol. 43,1947, pp.50-67.20
[8]J. Crank and P. Nicolson, A practical method for numerical integration of solutions of partial differential equations of heat-conduction type[J], Advances in Computational Mathematics, vol.6,1996, pp.207-226.20,109
[9]J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial value problems[J], J. Inst. Math. Applic., vol.18,1976, pp.189-202.20,55
[10]M. Van Daele and G. Vanden Berghe, P-stable Obrechkoff methods of arbitrary order for second-order differential equations[J], Numer. Algor., vol.44,2007, pp.115-131.20, 55,56
[11]Z. Wang, Y. Ge, Y. Dai, D. Zhao, A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.160,2004, pp.23-45.20,93,95
[12]D. Zhao, Z. Wang, Y. Dai, Importance of the first-order derivative formula in the Obrechkoff method[J], Comput. Phys. Commun., vol.167,2005, pp.65-75.21
[13]Z. Wang and Q. Chen, A trigonometrically-fitted one-step method with multi-derivative for the numerical solution to the one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.170,2005, pp.49-64.21,45
[14]A.J. Zakrzewski, Highly precise solutions of the one-dimensional Schroinger equation with an arbitrary potential[J], Comput. Phys. Commun., vol.175,2006, pp.397-403.8, 21
[15]冯康等编,数值计算方法[M],(国防工业出版社,1978),pp.473.24
[16]H. Shao and Z. Wang, Arbitrarily precise numerical solutions of the one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.180,2009, pp.1-7.37,38,41, 43,50,61,103
[17]J. Chen, Z. Wang, H. Shao, and H. Hao, Highly-accurate ground state energies of the He atom and the He-like ions by Hartree SCF calculation with Obrechkoff method[J], Comput. Phys. Commun., vol.179,2008, pp.486-491.38
[1]J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial value problems[J], J. Inst. Math. Applic., vol.18,1976, pp.189-202.20,55
[2]M. Van Daele and G. Vanden Berghe, P-stable Obrechkoff methods of arbitrary order for second-order differential equations[J], Numer. Algor., vol.44,2007, pp.115-131.20, 55,56
[3]U. Ananthakrishnaiah, P-stable Obrechkoff methods with minimal phase-lag for periodic initial value problems[J], Math. Comput. vol.49,1987, pp.553-559.55
[4]L. Brusa. and L. Nigro, A one-step method for direct integration of structural dynamic equations[J], Int. J. Numer. Meth. Engng., vol.15,1980, pp.553-559.56,102
[5]H. Shao and Z. Wang, Arbitrarily precise numerical solutions of the one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.180,2009, pp.1-7.37,38,41, 43,50,61,103
[6]P. M. Morse, Diatomic molecules according to the wave mechanics. Ⅱ. Vibrational lev-els[J], Phys. Rev., vol.34,1929, pp.57-64.63
[7]J. D. Praeger, Relaxational approach to solving the Schrodinger equation[J], Phys. Rev. A., vol.63,2001,022115.64
[8]S. Fliigge, Practical Quantum Mechanics[M]. (Berlin:Springer-Verlag,1974), pp.94. 65,105
[9]王竹溪,郭敦仁著,特殊函数概论[M],(北京大学出版社,2000),pp.131,137,288,289.64,66
[10]T. E. Stiefel and D. G. Bettis, Stabilization of Cowell's method[J], Numer. Math. vol. 13,1969, pp.154-175.68
[11]R. van Dooren, Stabilization of Cowell's classic finite difference method for numerical integration[J], J. Comput. Phys., vol.16,1974, pp.186-192.69
[12]Y. Dai, Z. Wang, D. Zhao, and X. Song, A new high efficient and high accurate Obrechkoff four-step method for the periodic nonlinear undamped duffing's equation[J], Comput. Phys. Commun., vol.165,2005, pp.110-126.70
[13]周纪卿,朱因远,非线性振动[M],(西安交通大学出版社,1998),pp.186-187.71
[14]D. Wu and Z. Wang, A Mathematica Program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach[J]. Computer Physics Communications, vol.174,2005, pp.447-463.75
[1]B. Numerov, A method of extrapolation of perturbations[J]. Monthly notices of the Royal Astronomical Society, vol.84,1924, pp.592-601.7,93
[2]Z. Wang, Y. Ge, Y. Dai, D. Zhao, A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.160,2004, pp.23-45.20,93,95
[3]L. Brusa. and L. Nigro, A one-step method for direct integration of structural dynamic equations[J], Int. J. Numer. Meth. Engng., vol.15,1980, pp.553-559.56,102
[4]H. Shao and Z. Wang, Arbitrarily precise numerical solutions of the one-dimensional Schroinger equation[J], Comput. Phys. Commun., vol.180,2009, pp.1-7.37,38,41, 43,50,61,103
[5]S. Fliigge, Practical Quantum Mechanics[M]. (Berlin:Springer-Verlag,1974), pp.94. 65,105
[1]J. Crank and P. Nicolson, A practical method for numerical integration of solutions of partial differential equations of heat-conduction type[J], Advances in Computational Mathematics, vol.6,1996, pp.207-226.20,109
[2]W. van Dijk and F. M. Toyama, Accurate numerical solutions of the time-dependent Schroinger equation[J], Phys. Rev. E, vol.75,2007,036707.3,109,118
[3]T.Iitaka, Solving the time-dependent Schrodinger equation numerically[J]. Phys. Rev. E, vol.49,1994, pp.4684-4690.109
[4]I. Puzynin, A. Selin, and S. Vinitsky, A high-order accuracy method for numerical solv-ing of the time-dependent Schroinger equation[J], Comput. Phys. Commun., vol.123, 1999, pp.1-6.109,118
[5]I. Puzynin, A. Selin, and S. Vinitsky, Magnus-factorized method for numerical solving the time-dependent Schrodinger equation[J], Comput. Phys. Commun., vol.126,2000, pp.158-161.109
[6](?). Mi(?)icu, M. Rizea, and W. Greiner, Emission of electromagnetic radiation in alpha decay[J], J. Phys. G, vol.27,2001, pp.993-1003.109
[7]W. S. Dias, E. M. Nascimento, M. L. Lyra, and F. A. B. F. de Moura, Frequency doubling of Bloch oscillations for interacting electrons in a static electric field[J], Phys. Rev. B, vol.76,2007,155124.3,109
[8]S. Wolfram, The Mathematica Book,5th ed[M]. (Wolfram Media, Inc.2003).
[9]A. Goldberg, H. M. Schey, and J. L. Schwartz, Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena[J], Am. J. Phys., vol.35,1967, pp.177-186.118
[10]L. I. Schiff, Quantum Mechanics, International Series in Pure and Applied Physics,1st ed[M]. (McGraw-Hill Book Company Inc., New York,1949), pp:67.118,119,121
[11]T. Cheon, I. Tsutsui, and T. Fulop, Quantum Abacus[J], Phys. Lett. A, vol.330,2004, pp:338-342.119