High algebraic order methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation

Journal of Mathematical Chemistry

Volumn 48, Issue 4, 2010, Pages 925-958

  Alolyan, Ibraheem ^a   Simos, T E ^a,b  

^a KING SAUD UNIVERSITY   (Saudi Arabia)

^b UNIVERSITY OF PELOPONNESE   (Greece)

Author keywords

Hybrid methods; Interval of periodicity; Multistep methods; Numerical solution; P stability; Phase fitted; Phase lag; Schr dinger equation

Indexed keywords

EID: 77957352499     PISSN: 02599791     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10910-010-9718-y     Document Type: Article

References (107)

1. Ixaru L. Gr., Micu M.: Topics in Theoretical Physics Central. Institute of Physics, Bucharest (1978).
2. Landau L. D., Lifshitz F. M.: Quantum Mechanics. Pergamon, New York (1965).
3. Prigogine I.: Stuart Rice (Eds): Advances in Chemical Physics Vol. 93: New Methods in Computational Quantum Mechanics. Wiley, New York (1997).
4. Herzberg G.: Spectra of Diatomic Molecules. van Nostrand, Toronto (1950).
5. T. E. Simos, Atomic structure computations in chemical modelling: applications and theory (Editor: A. Hinchliffe, UMIST). R. Soc. Chem. 38-142(2000).
6. Simos T. E.: Numerical methods for 1D, 2D and 3D differential equations arising in chemical problems, chemical modelling: application and theory. R. Soc. Chem. 2, 170-270 (2002).
7. T. E. Simos: Numerical Solution of Ordinary Differential Equations with Periodical Solution. Doctoral Dissertation, National Technical University of Athens, Greece, 1990.
8. Dormand J. R., El-Mikkawy M. E. A., Prince P. J.: Families of Runge-Kutta-Nyström formulae. IMA J. Numer. Anal. 7, 235-250 (1987).
9. Sideridis A. B., Simos T. E.: A low-order embedded Runge-Kutta method for periodic initial-value problems. J. Comput. Appl. Math. 44(2), 235-244 (1992).
10. Simos T. E.: A Runge-Kutta Fehlberg method with phase-lag of order infinity for initial value problems with oscillating solution. Comput. Math. Appl. 25, 95-101 (1993).
11. Simos T. E.: Runge-Kutta interpolants with minimal phase-lag. Comput. Math. Appl. 26, 43-49 (1993).
12. Simos T. E.: Runge-Kutta-Nyström interpolants for the numerical integration of special second-order periodic initial-value problems. Comput. Math. Appl. 26, 7-15 (1993).
13. Simos T. E.: A high-order predictor-corrector method for periodic IVPs. Appl. Math. Lett. 6(5), 9-12 Sep (1993).
14. Simos T. E., Dimas E., Sideridis A. B.: A Runge-Kutta-Nyström method for the numerical-integration of special 2nd-order periodic initial-value problems. J. Comput. Appl. Math. 51(3), 317-326 (1994).
15. Simos T. E.: An explicit high-order predictor-corrector method for periodic initial-value problems. Math. Models & Methods Appl. Sci. 5(2), 159-166 (1995).
16. Avdelas G., Simos T. E.: Block Runge-Kutta methods for periodic initial-value problems. Comput. Math. Appl. 31, 69-83 (1996).
17. Avdelas G., Simos T. E.: Embedded methods for the numerical solution of the Schrödinger equation. Comput. Math. Appl. 31, 85-102 (1996).
18. Simos T. E.: A modified Runge-Kutta method for the numerical solution of ODE's with oscillation solutions. Appl. Math. Lett. 9(6), 61-66 (1996).
19. Simos T. E.: Some embedded modified Runge-Kutta methods for the numerical solution of some specific Schrödinger equations. J. Math. Chem. 24(1-3), 23-37 (1998).
20. Simos T. E.: An embedded Runge-Kutta method with phase-lag of order infinity for the numerical solution of the of Schrödinger equation. Int. J. Mod. Phys. C 11, 1115-1133 (2000).
21. Simos T. E., Vigo-Aguiar Jesus: A new modified Runge-Kutta-Nyström method with phase-lag of order infinity for the numerical solution of the Schrödinger equation and related problems. Int. J. Mod. Phys. C 11, 1195-1208 (2000).
22. Simos T. E., Vigo-Aguiar Jesus: A modified Runge-Kutta method with phase-lag of order infinity for the numerical solution of the of Schrödinger equation and related problems. Comput. Chem. 25, 275-281 (2001).
23. Simos T. E., Vigo-Aguiar J.: A modified phase-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation. J. Math. Chem. 30(1), 121-131 (2001).
24. Simos T. E., Williams P. S.: A new Runge-Kutta-Nyström method with phase-lag of order infinity for the numerical solution of the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 45, 123-137 (2002).
25. Tsitouras Ch., Simos T. E.: Optimized Runge-Kutta pairs for problems with oscillating solutions. J. Comput. Appl. Math. 147(2), 397-409 (2002).
26. Dormand J. R., Prince P. J.: A family of embedded RungeKutta formulae. J. Comput. Appl. Math. 6, 1926 (1980).
27. Anastassi Z. A., Simos T. E.: Special optimized Runge-Kutta methods for IVPs with oscillating solutions. Int. J. Mod. Phys. C 15, 1-15 (2004).
28. Anastassi Z. A., Simos T. E.: A dispersive-fitted and dissipative-fitted explicit Runge-Kutta method for the numerical solution of orbital problems. New Astron. 10, 31-37 (2004).
29. Tselios K., Simos T. E.: Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. J. Comput. Appl. Math. 175(1), 173-181 (2005).
30. Anastassi Z. A., Simos T. E.: An optimized Runge-Kutta method for the solution of orbital problems. J. Comput. Appl. Math. 175(1), 1-9 (2005).
31. Triantafyllidis T. V., Anastassi Z. A., Simos T. E.: Two optimized Runge-Kutta methods for the solution of the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 60(3), 753-771 (2008).
32. Papadopoulos D. F, Anastassi Z. A., Simos T. E.: A phase-fitted Runge-Kutta-Nystrom method for the numerical solution of initial value problems with oscillating solutions. Comput. Phys. Commun. 180(10), 1839-1846 (2009).
33. Kosti A. A., Anastassi Z. A., Simos T. E.: An optimized explicit Runge-Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem. 47(1), 315-330 (2010).
34. Lambert J. D., Watson I. A.: Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189-202 (1976).
35. Raptis A. D., Allison A. C.: Exponential-fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 14, 1-5 (1978).
36. Raptis A. D.: Exponentially-fitted solutions of the eigenvalue Shrödinger equation with automatic error control. Comput. Phys. Commun. 28, 427-431 (1983).
37. Kalogiratou Zacharoula, Simos T. E.: A P-stable exponentially-fitted method for the numerical integration of the Schrödinger equation. Appl. Math. Comput. 112, 99-112 (2000).
38. Raptis A. D., Simos T. E.: A four-step phase-fitted method for the numerical integration of second order initial-value problem. BIT 31, 160-168 (1991).
39. Simos T. E., Raptis A. D.: Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrödinger equation. Computing 45, 175-181 (1990).
40. Simos T. E.: A two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problems. Int. J. Comput. Math. 39, 135-140 (1991).
41. Simos T. E.: A Numerov-type method for the numerical-solution of the radial Schrödinger-equation. Appl. Numer. Math. 7(2), 201-206 (1991).
42. Simos T. E.: Explicit two-step methods with minimal phase-lag for the numerical-integration of special second-order initial-value problems and their application to the one-dimensional Schrödinger-equation. J. Comput. Appl. Math. 39(1), 89-94 (1992).
43. Simos T. E.: Two-step almost P-stable complete in phase methods for the numerical integration of second order periodic initial-value problems. Int. J. Comput. Math. 46, 77-85 (1992).
44. Simos T. E.: An explicit almost P-stable two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problems. Appl. Math. Comput. 49, 261-268 (1992).
45. Simos T. E.: High-order methods with minimal phase-lag for the numerical integration of the special second order initial value problem and their application to the one-dimensional Schrödinger equation. Comput. Phys. Commun. 74, 63-66 (1993).
46. Simos T. E.: A new variable-step method for the numerical-integration Of special 2Nd-order initial-value problems and their application to the one-dimensional SchrÖdinger-equation. Appl. Math. Lett. 6(3), 67-73 (1993).
47. Simos T. E.: A family of two-step almost P-stable methods with phase-lag of order infinity for the numerical integration of second order periodic initial-value problems. Jpn. J. Ind. Appl. Math. 10, 289-297 (1993).
48. Simos T. E.: A predictor-corrector phase-fitted method for y′′ = f(x, y). Math. Comput. Simul. 35, 153-159 (1993).
49. Simos T. E.: A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial value problems. Proc. R. Soc. Lond. Ser. A 441, 283-289 (1993).
50. Simos T. E.: An explicit 4-step phase-fitted method for the numerical-integration of 2nd-order initial-value problems. J. Comput. Appl. Math. 55(2), 125-133 (1994).
51. Simos T. E.: Some new variable-step methods with minimal phase-lag for the numerical integration of special 2nd-order initial value problems. Appl. Math. Comput. 64, 65-72 (1994).
52. Simos T. E., Mousadis G.: Some new Numerov-type methods with minimal phase-lag for the numerical integration of the radial Schrödinger equation. Mole. Phys. 83, 1145-1153 (1994).
53. Simos T. E., Mousadis G.: A two-step method for the numerical solution of the radial Schrdinger equation. Comput. Math. Appl. 29, 31-37 (1995).
54. Simos T. E.: Predictor corrector phase-fitted methods for Y′′ = F(X, Y) and an application to the Schrödinger-equation. Int. J. Quantum Chem. 53(5), 473-483 (1995).
55. Simos T. E.: Some low-order two-step almost P-stable methods with phase-lag of order infinity for the numerical integration of the radial Schrödinger equation. Int. J. Mod. Phys. A 10, 2431-2438 (1995).
56. Simos T. E.: A new Numerov-type method for computing eigenvalues and resonances of the radial Schrödinger equation. Int. J. Mod. Phys. C-Phys. Comput. 7(1), 33-41 (1996).
57. Papakaliatakis G., Simos T. E.: A new method for the numerical solution of fourth order BVPs with oscillating solutions. Comput. Math. Appl. 32, 1-6 (1996).
58. Simos T. E.: Accurate computations for the elastic scattering phase-shift problem. Comput. Chem. 21, 125-128 (1996).
59. Simos T. E.: An eighth order method with minimal phase-lag for accuarate computations for the elastic scattering phase-shift problem. Int. J. Mod. Phys. C 7, 825-835 (1996).
60. Simos T. E.: Eighth order methods with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem. J. Math. Chem. 21(4), 359-372 (1997).
61. Simos T. E.: An extended Numerov-type method for the numerical solution of the Schrödinger equation. Comput. Math. Appl. 33, 67-78 (1997).
62. Simos T. E.: New Numerov-type methods for computing eigenvalues, resonances and phase shifts of the radial Schrödinger equation. Int. J. Quantum Chem. 62, 467-475 (1997).
63. Simos T. E.: New P-stable high-order methods with minimal phase-lag for the numerical integration of the radial Schrödinger equation. Phys. Scr. 55, 644-650 (1997).
64. Simos T. E.: Eighth order methods for elastic scattering phase-shifts. Int. J. Theor. Phys. 36, 663-672 (1997).
65. Simos T. E., Tougelidis G.: An explicit eighth order method with minimal phase-lag for the numerical solution of the Schrödinger equation. Comput. Mater. Sci. 8, 317-326 (1997).
66. Simos T. E., Tougelidis G.: An explicit eighth order method with minimal phase-lag for accurate computations of eigenvalues, resonances and phase shifts. Comput. Chem. 21, 327-334 (1997).
67. Simos T. E.: Eighth order method for accurate computations for the elastic scattering phase-shift problem. Int. J. Quantum Chem. 68, 191-200 (1998).
68. Simos T. E.: New embedded explicit methods with minimal phase-lag for the numerical integration of the Schrödinger equation. Comp. Chem. 22, 433-440 (1998).
69. Simos T. E.: High-algebraic, high-phase-lag methods for accurate computations for the elastic-scattering phase shift problem. Can. J. Phys. 76, 473-493 (1998).
70. Simos T. E.: High algebraic order methods with minimal phase-lag for accurate solution of the Schrödinger equation. Int. J. Mod. Phys. C 9, 1055-1071 (1998).
71. Avdelas G., Simos T. E.: Embedded eighth order methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 26(4), 327-341 (1999).
72. Simos T. E.: A new finite difference scheme with minimal phase-lag for the numerical solution of the Schrödinger equation. Appl. Math. Comput. 106, 245-264 (1999).
73. Simos T. E.: High algebraic order explicit methods with reduced phase-lag for an efficient solution of the Schrödinger equation. Int. J. Quantum Chem. 73, 479-496 (1999).
74. Simos T. E.: Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrd̈inger equation. Comput. Chem. 23, 439-446 (1999).
75. Simos T. E.: Explicit eighth order methods for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. Phys. Commun. 119, 32-44 (1999).
76. Simos T. E.: High algebraic order methods for the numerical solution of the Schrödinger equation. Mole. Simul. 22, 303-349 (1999).
77. Avdelas G., Konguetsof A., Simos T. E.: A family of hybrid eighth order methods with minimal phase-lag for the numerical solution of the Schrödinger equation and related problems. Int. J. Mod. Phys. C 11, 415-437 (2000).
78. Avdelas G., Simos T. E.: Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation. Phys. Rev. E 62, 1375-1381 (2000).
79. Avdelas G., Simos T. E.: On variable-step methods for the numerical solution of Schrödinger equation and related problems. Comput. Chem. 25, 3-13 (2001).
80. Simos T. E., Williams P. S.: New insights in the development of Numerov-type methods with minimal phase-lag for the numerical solution of the Schrödinger equation. Comput. Chem. 25, 77-82 (2001).
81. Avdelas G., Konguetsof A., Simos T. E.: A generator of hybrid explicit methods for the numerical solution of the Schrödinger equation and related problems. Comput. Phys. Commun. 136, 14-28 (2001).
82. Simos T. E., Vigo-Aguiar J.: A symmetric high-order method with minimal phase-lag for the numerical solution of the Schrödinger equation. Int. J. Mod. Phys. C 12, 1035-1042 (2001).
83. Simos T. E., Vigo-Aguiar J.: On the construction of efficient methods for second order IVPs with oscillating solution. Int. J. Mod. Phys. C 12, 1453-1476 (2001).
84. Avdelas G., Konguetsof A., Simos T. E.: A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation Part 1. Development of the basic method. J. Math. Chem. 29(4), 281-291 (2001).
85. Avdelas G., Konguetsof A., Simos T. E.: A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation. Part 2. Development of the generator; optimization of the generator and numerical results. J. Math. Chem 29(4), 293-305 (2001).
86. Tsitouras Ch., Simos T. E.: High algebraic, high phase-lag order embedded Numerov-type methods for oscillatory problems. Appl. Math. Comput. 131, 201-211 (2002).
87. Avdelas G., Konguetsof A., Simos T. E.: A generator of dissipative methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 148, 59-73 (2002).
88. Konguetsof A., Simos T. E.: P-stable eighth algebraic order methods for the numerical solution of the Schrödinger equation. Comput. Chem. 26, 105-111 (2002).
89. Simos T. E., Vigo-Aguiar J.: Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrödinger equation. J. Math. Chem. 31(2), 135-144 (2002).
90. Konguetsof A., Simos T. E.: A generator of hybrid symmetric four-step methods for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 158(1), 93-106 (2003).
91. Simos T. E., Famelis I. T., Tsitouras Ch.: Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions. Numer. Algorithms 34(1), 27-40 (2003).
92. Sakas D. P., Simos T. E.: Multiderivative methods of eighth algrebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation. J. Comput. Appl. Math. 175(1), 161-172 (2005).
93. Sakas D. P., Simos T. E.: A family of multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 317-331 (2005).
94. Panopoulos G. A., Anastassi Z. A., Simos T. E.: Two new optimized eight-step symmetric methods for the efficient solution of the Schrödinger equation and related problems. MATCH Commun. Math. Comput. Chem. 60(3), 773-785 (2008).
95. Simos T. E.: A new Numerov-type method for the numerical solution of the Schrödinger equation. J. Math. Chem. 46(3), 981-1007 (2009).
96. Konguetsof A.: A new two-step hybrid method for the numerical solution of the Schrödinger equation. J. Math. Chem. 47(2), 871-890 (2010).
97. Simos T. E., Williams P. S.: On finite difference methods for the solution of the Schrödinger equation. Comput. Chem. 23, 513-554 (1999).
98. Anastassi Z. A., Simos T. E.: Numerical multistep methods for the efficient solution of quantum mechanics and related problems. Phys. Rep.-Rev. Sect. Phys. Lett. 482, 1-240 (2009).
99. Vigo-Aguiar J., Simos T. E.: Review of multistep methods for the numerical solution of the radial Schrödinger equation. Int. J. Quantum Chem. 103(3), 278-290 (2005).
100. Simos T. E., Zdetsis A. D., Psihoyios G., Anastassi Z. A.: Special issue on mathematical chemistry based on papers presented within ICCMSE 2005 preface. J. Math. Chem. 46(3), 727-728 (2009).
101. Simos T. E., Psihoyios G.: Special issue: the international conference on computational methods in sciences and engineering 2004-preface. J. Comput. Appl. Math. 191(2), 165-165 (2006).
102. Simos T. E., Psihoyios G.: Special issue-selected papers of the international conference on computational methods in sciences and engineering (ICCMSE 2003) Kastoria, Greece, 12-16 September 2003-Preface. J. Comput. Appl. Math. 175(1), IX-IX (2005).
103. Simos T. E., Vigo-Aguiar J.: Special issue-selected papers from the conference on computational and mathematical methods for science and engineering (CMMSE-2002)-Alicante University, Spain, 20-25 September 2002-preface. J. Comput. Appl. Math. 158(1), IX-IX (2003).
104. Simos T. E., Williams P. S.: A finite-difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 79(2), 189-205 (1997).
105. Ixaru L. Gr., Rizea M.: Comparison of some four-step methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 38(3), 329-337 (1985).
106. Ixaru L. Gr., Rizea M.: A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Comput. Phys. Commun. 19, 23-27 (1980).
107. Quinlan G. D., Tremaine S.: Astron. J. 100(5), 1694-1700 (1990).