High order closed Newton-Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation

Applied Mathematics and Computation

Volumn 209, Issue 1, 2009, Pages 137-151

  Simos, T E ^a  

^a UNIVERSITY OF PELOPONNESE   (Greece)

Author keywords

Closed Newton Cotes differential methods; Energy preservation; Error analysis; Multistep methods; Numerical methods; Orbital problems; Schr dinger equation; Symplectic integrators; Trigonometric fitting

Indexed keywords

BEAM PLASMA INTERACTIONS; ERROR ANALYSIS; NUMBER THEORY;

CLOSED NEWTON-COTES DIFFERENTIAL METHODS; ENERGY PRESERVATION; MULTISTEP METHODS; ORBITAL PROBLEMS; SYMPLECTIC INTEGRATORS; TRIGONOMETRIC FITTING;

NUMERICAL METHODS;

EID: 60549092695     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2008.06.020     Document Type: Article

References (122)

1. Zhu W., Zhao X., and Tang Y. Numerical methods with a high order of accuracy in the quantum system. Journal of Chemical Physics 104 (1996) 2275-2286
2. Chiou J.C., and Wu S.D. Open Newton-Cotes differential methods as multilayer symplectic integrators. Journal of Chemical Physics 107 (1997) 6894-6897
3. Sanz-Serna J.M., and Calvo M.P. Numerical Hamiltonian Problem (1994), Chapman and Hall, London
4. Simos T.E. Atomic structure computations in chemical modelling: applications and theory. In: Hinchliffe A. (Ed). UMIST (2000), The Royal Society of Chemistry 38-142
5. Simos T.E. Numerical methods for 1D, 2D and 3D differential equations arising in chemical problems. Chemical Modelling: Application and Theory vol. 2 (2002), The Royal Society of Chemistry pp. 170-270
6. Simos T.E., and Williams P.S. On finite difference methods for the solution of the Schrödinger equation. Computers and Chemistry 23 (1999) 513-554
7. T.E. Simos, Numerical Solution of Ordinary Differential Equations with Periodical Solution, Doctoral Dissertation, National Technical University of Athens, Greece, 1990 (in Greek).
8. Raptis A.D. Exponential multistep methods for ordinary differential equations. Bulletin of the Greek Mathematical Society 25 (1984) 113-126
9. Simos T.E., and Williams P.S. A family of Numerov-type exponentially-fitted methods for the numerical integration of the Schrödinger equation. Computers and Chemistry 21 (1997) 403-417
10. Thomas R.M., and Simos T.E. A family of hybrid exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation. Journal of Computational and Applied Mathematics 87 (1997) 215-226
11. Cash J.R., Raptis A.D., and Simos T.E. A sixth order exponentially-fitted method for the numerical solution of the radial Schrödinger equation. Journal of Computational Physics 91 (1990) 413-423
12. Liu X.S., Liu X.Y., Zhou Z.Y., Ding P.Z., and Pan S.F. Numerical solution of one-dimensional time-independent Schrödinger equation by using symplectic schemes. International Journal of Quantum Chemistry 79 (2000) 343-349
13. L.Gr. Ixaru, Numerical Methods for Differential Equations and Applications, Reidel, Dordrecht, Boston, Lancaster, 1984.
14. Ixaru L.Gr., and Rizea M. A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Computer Physics Communications 19 (1980) 23-27
15. Avdelas G., and Simos T.E. Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation. Physical Review E 62 (2000) 1375-1381
16. Avdelas G., and Simos T.E. Embedded methods for the numerical solution of the Schrödinger equation. Computer and Mathematics with Applications 31 (1996) 85-102
17. Avdelas G., and Simos T.E. A generator of high-order embedded P-stable method for the numerical solution of the Schrödinger equation. Journal of Computational and Applied Mathematics 72 (1996) 345-358
18. ″ = f (x, y) and an application to the Schrödinger equation. International Journal of Quantum Chemistry 53 (1995) 473-483
19. Simos T.E. Eighth-order method for accurate computations for the elastic scattering phase-shift problem. International Journal of Quantum Chemistry 68 (1998) 191-200
20. Raptis A.D., and Allison A.C. Exponential-fitting methods for the numerical solution of the Schrödinger equation. Computer Physics Communications 14 (1978) 1-5
21. Simos T.E. An eighth order exponentially-fitted method for the numerical solution of the Schrödinger equation. International Journal of Modern Physics C 9 (1998) 271-288
22. Avdelas G., Konguetsof A., and 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. International Journal of Modern Physics C 11 (2000) 415-437
23. Simos T.E., and Williams P.S. A new Runge-Kutta-Nyström method with phase-lag of order infinity for the numerical solution of the Schrödinger equation. MATCH Communications in Mathematical and in Computer Chemistry 45 (2002) 123-137
24. Konguetsof A., and Simos T.E. On the construction of exponentially-fitted methods for the numerical solution of the Schrödinger equation. Journal of Computational Methods in Sciences and Engineering 1 (2001) 143-165
25. Simos T.E. P-stable four-step exponentially-fitted method for the numerical integration of the Schrödinger equation. Computing Letters 1 1 (2005) 37-45
26. Enright W.H. On the use of 'arc length' and 'defect' for mesh selection for differential equations. Computing Letters 1 2 (2005) 47-52
27. Psihoyios G. A block implicit advanced step-point (BIAS) algorithm for stiff differential systems. Computing Letters 2 1-2 (2006) 51-58
28. Simos T.E. A new Numerov-type method for computing eigenvalues and resonances of the radial Schrödinger equation. International Journal of Modern Physics C - Physics and Computers 7 1 (1996) 33-41
29. Simos T.E., and Mousadis G. Some new Numerov-type methods with minimal phase-lag for the numerical-integration of the radial Schrödinger equation. Molecular Physics 83 6 (1994) 1145-1153
30. Simos T.E. A Numerov-type method for the numerical-solution of the radial Schrödinger equation. Applied Numerical Mathematics 7 2 (1991) 201-206
31. Simos T.E. High algebraic order methods with minimal phase-lag for accurate solution of the Schrödinger equation. International Journal of Modern Physics C 9 7 (1998) 1055-1071
32. Simos T.E., and Williams P.S. Bessel and Neumann-fitted methods for the numerical solution of the radial Schrödinger equation. Computers and Chemistry 21 3 (1997) 175-179
33. Anastassi Z.A., and Simos T.E. A family of exponentially-fitted Runge-Kutta methods with exponential order up to three for the numerical solution of the Schrödinger equation. Journal of Mathematical Chemistry 41 1 (2007) 79-100
34. Monovasilis T., Kalogiratou Z., and Simos T.E. Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. Journal of Mathematical Chemistry 40 3 (2006) 257-267
35. Psihoyios G., and Simos T.E. The numerical solution of the radial Schrödinger equation via a trigonometrically fitted family of seventh algebraic order predictor-corrector methods. Journal of Mathematical Chemistry 40 3 (2006) 269-293
36. Simos T.E. A four-step exponentially fitted method for the numerical solution of the Schrödinger equation. Journal of Mathematical Chemistry 40 3 (2006) 305-318
37. Monovasilis T., Kalogiratou Z., and Simos T.E. Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. Journal of Mathematical Chemistry 37 3 (2005) 263-270
38. Kalogiratou Z., Monovasilis T., and Simos T.E. Numerical solution of the two-dimensional time independent Schrödinger equation with Numerov-type methods. Journal of Mathematical Chemistry 37 3 (2005) 271-279
39. Anastassi Z.A., and Simos T.E. Trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation. Journal of Mathematical Chemistry 37 3 (2005) 281-293
40. Psihoyios G., and Simos T.E. Sixth algebraic order trigonometrically fitted predictor-corrector methods for the numerical solution of the radial Schrödinger equation. Journal of Mathematical Chemistry 37 3 (2005) 295-316
41. Sakas D.P., and Simos T.E. A family of multiderivative methods for the numerical solution of the Schrödinger equation. Journal of Mathematical Chemistry 37 3 (2005) 317-331
42. Simos T.E. Exponentially-fitted multiderivative methods for the numerical solution of the Schrödinger equation. Journal of Mathematical Chemistry 36 1 (2004) 13-27
43. Tselios K., and Simos T.E. Symplectic methods of fifth order for the numerical solution of the radial Shrodinger equation. Journal of Mathematical Chemistry 35 1 (2004) 55-63
44. Simos T.E. A family of trigonometrically-fitted symmetric methods for the efficient solution of the Schrödinger equation and related problems. Journal of Mathematical Chemistry 34 1-2 (2003) 39-58
45. Tselios K., and Simos T.E. Symplectic methods for the numerical solution of the radial Shrödinger equation. Journal of Mathematical Chemistry 34 1-2 (2003) 83-94
46. Vigo-Aguiar J., and Simos T.E. Family of twelve steps exponential fitting symmetric multistep methods for the numerical solution of the Schrödinger equation. Journal of Mathematical Chemistry 32 3 (2002) 257-270
47. Avdelas G., Kefalidis E., and Simos T.E. New P-stable eighth algebraic order exponentially-fitted methods for the numerical integration of the Schrödinger equation. Journal of Mathematical Chemistry 31 4 (2002) 371-404
48. Simos T.E., and Vigo-Aguiar J. Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrödinger equation. Journal of Mathematical Chemistry 31 2 (2002) 135-144
49. Z. Kalogiratou, T.E. Simos, Construction of trigonometrically and exponentially fitted Runge-Kutta-Nyström methods for the numerical solution of the Schrödinger equation and related problems a method of eighth algebraic order, Journal of Mathematical Chemistry 31 (2) 211-232.
50. Simos T.E., and Vigo-Aguiar J. A modified phase-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation. Journal of Mathematical Chemistry 30 1 (2001) 121-131
51. Avdelas G., Konguetsof A., and 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. Journal of Mathematical Chemistry 29 4 (2001) 281-291
52. Avdelas G., Konguetsof A., and 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. Journal of Mathematical Chemistry 29 4 (2001) 293-305
53. Vigo-Aguiar J., and Simos T.E. A family of P-stable eighth algebraic order methods with exponential fitting facilities. Journal of Mathematical Chemistry 29 3 (2001) 177-189
54. Simos T.E. A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation. Journal of Mathematical Chemistry 27 4 (2000) 343-356
55. Avdelas G., and Simos T.E. Embedded eighth order methods for the numerical solution of the Schrödinger equation. Journal of Mathematical Chemistry 26 4 (1999) 327-341
56. Simos T.E. A family of P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation. Journal of Mathematical Chemistry 25 1 (1999) 65-84
57. Simos T.E. Some embedded modified Runge-Kutta methods for the numerical solution of some specific Schrödinger equations. Journal of Mathematical Chemistry 24 1-3 (1998) 23-37
58. Simos T.E. Eighth order methods with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem. Journal of Mathematical Chemistry 21 4 (1997) 359-372
59. Amodio P., Gladwell I., and Romanazzi G. Numerical solution of general bordered ABD linear systems by cyclic reduction. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 1 1 (2006) 5-12
60. Capper S.D., Cash J.R., and Moore D.R. Lobatto-Obrechkoff formulae for 2nd order two-point boundary value problems. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 1 1 (2006) 13-25
61. Capper S.D., and Moore D.R. On high order MIRK schemes and Hermite-Birkhoff interpolants. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 1 1 (2006) 7-47
62. Cash J.R., Sumarti N., Abdulla T.J., and Vieira I. The derivation of interpolants for nonlinear two-point boundary value problems. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 1 1 (2006) 49-58
63. Cash J.R., and Girdlestone S. Variable step Runge-Kutta-Nyström methods for the numerical solution of reversible systems. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 1 1 (2006) 59-80
64. Cash J.R., and Mazzia F. Hybrid mesh selection algorithms based on conditioning for two-point boundary value problems. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 1 1 (2006) 81-90
65. Iavernaro F., Mazzia F., and Trigiante D. Stability and conditioning in numerical analysis. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 1 1 (2006) 91-112
66. Iavernaro F., and Trigiante D. Discrete conservative vector fields induced by the trapezoidal method. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 1 1 (2006) 113-130
67. Mazzia F., Sestini A., and Trigiante D. BS linear multistep methods on non-uniform meshes. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 1 1 (2006) 131-144
68. Shampine L.F., Muir P.H., and Xu H. A user-friendly Fortran BVP solver. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 1 2 (2006) 201-217
69. Vanden Berghe G., and Van Daele M. Exponentially-fitted Strmer/Verlet methods. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 1 3 (2006) 241-255
70. Aceto L., Pandolfi R., and Trigiante D. Stability analysis of linear multistep methods via polynomial type variation. JNAIAM Journal of Numerical Analysis, Industrial, and Applied Mathematics 2 1-2 (2007) 1-9
71. Lambert J.D., and Watson I.A. Symmetric multistep methods for periodic initial values problems. Journal of the Institute of Mathematics and its Applications 18 (1976) 189-202
72. Van Der Houwen P.J., and Sommeijer B.P. Explicit Runge-Kutta-Nyström methods with reduced phase errors for computing oscillating solutions. SIAM Journal of Numerical Analysis 24 (1987)
73. ″ = f (x, y), in: D. Bainov, V. Civachev (Eds.), Proceedings of the First International Colloquium on Numerical Analysis, Bulgaria, 1992, pp. 27-38.
74. ″ = f (x, y) via rational approximation for the cosine. IMA Journal of Numerical Analysis 9 (1989) 145-165
75. Raptis A.D., and Simos T.E. A four-step phase-fitted method for the numerical integration of second order initial-value problems. BIT 31 (1991) 160-168
76. Chawla M.M. Numerov made explicit has better stability. BIT 24 (1984) 117-118
77. Chawla M.M., and Rao P.S. A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II. Explicit method. Journal of Computational and Applied Mathematics 15 (1986) 329-337
78. Blatt J.M. Practical points concerning the solution of the Schrödinger equation. Journal of Computational Physics 1 (1967) 382-396
79. Cooley J.W. An improved eigenvalue corrector formula for solving Schrödinger's equation for central fields. Mathematics of Computation 15 (1961) 363-374
80. Dormand J.R., and Prince P.J. A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics 6 (1980) 19-26
81. Prince P.J., and Dormand J.R. High order embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics 7 (1981) 67-75
82. Simos T.E. Multiderivative methods for the numerical solution of the Schroedinger equation. MATCH Communications in Mathematical and in Computer Chemistry 50 (2004) 7-26
83. Sakas D.P., and Simos T.E. Trigonometrically-fitted multiderivative methods for the numerical solution of the radial Schroedinger equation. MATCH Communications in Mathematical and in Computer Chemistry 53 2 (2005) 299-320
84. Psihoyios G., and Simos T.E. A family of fifth algebraic order trigonometrically fitted P-C schemes for the numerical solution of the radial Schroedinger equation. MATCH Communications in Mathematical and in Computer Chemistry 53 2 (2005) 321-334
85. Kalogiratou Z., and Simos T.E. A P-stable exponentially fitted method for the numerical integration of the Schrödinger equation. Applied Mathematics and Computation 112 (2000) 99-112
86. 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 initialvalue problems. Applied Mathematics and Computation 49 (1992) 261-268
87. Simos T.E. Some new variable-step methods with minimal phase-lag for the numerical integration of special 2nd-order initial value problems. Applied Mathematics and Computation 64 (1994) 65-72
88. Simos T.E. Modified Runge-Kutta methods for the numerical solution of ODEs with oscillating solutions. Applied Mathematics and Computation 84 (1997) 131-143
89. Tsitouras Ch., and Simos T.E. Explicit high order methods for the numerical integration of periodic initial-value problems. Applied Mathematics and Computation 95 (1998) 15-26
90. Simos T.E. Explicit exponentially-fitted methods for the numerical solution of the Schrdinger equation. Applied Mathematics and Computation 98 (1999) 199-208
91. Simos T.E. A new finite difference scheme with minimal phase-lag for the numerical solution of the Schrodinger equation. Applied Mathematics and Computation 106 (1999) 245-264
92. Tsitouras Ch., and Simos T.E. High algebraic, high phase-lag order embedded Numerov-type methods for oscillatory problems. Applied Mathematics and Computation 131 (2002) 201-211
93. Simos T.E. A Runge-Kutta Fehlberg method with phase-lag of order infinity for initial value problems with oscillating solution. Computers and Mathematics with Applications 25 (1993) 95-101
94. Simos T.E. Runge-Kutta interpolants with minimal phase-lag. Computers and Mathematics with Applications 26 (1993) 43-49
95. Simos T.E. Runge-Kutta-Nyström interpolants for the numerical integration of special second-order periodic initial-value problems. Computers and Mathematics with Applications 26 (1993) 7-15
96. Simos T.E., and Mitsou G.V. A family of four-step exponential fitted methods for the numerical integration of the radial Schrödinger equation. Computers and Mathematics with Applications 28 (1994) 41-50
97. Simos T.E., and Mousadis G. A two-step method for the numerical solution of the radial Schrdinger equation. Computers and Mathematics with Applications 29 (1995) 31-37
98. Avdelas G., and Simos T.E. Block Runge-Kutta methods for periodic initial-value problems. Computers and Mathematics with Applications 31 (1996) 69-83
99. Avdelas G., and Simos T.E. Embedded methods for the numerical solution of the Schrödinger equation. Computers and Mathematics with Applications 31 (1996) 85-102
100. Papakaliatakis G., and Simos T.E. A new method for the numerical solution of fourth order BVPs with oscillating solutions. Computers and Mathematics with Applications 32 (1996) 1-6
101. Simos T.E. An extended Numerov-type method for the numerical solution of the Schrödinger equation. Computers and Mathematics with Applications 33 (1997) 67-78
102. Simos T.E. A new hybrid imbedded variable-step procedure for the numerical integration of the Schrödinger equation. Computers and Mathematics with Applications 36 (1998) 51-63
103. Simos T.E. Bessel and Neumann fitted methods for the numerical solution of the Schrödinger equation. Computers and Mathematics with Applications 42 (2001) 833-847
104. Konguetsof A., and Simos T.E. An exponentially-fitted and trigonometrically-fitted method for the numerical solution of periodic initial-value problems. Computers and Mathematics with Applications 45 (2003) 547-554
105. Simos T.E. A new variable step method for the numerical integration of special second order initial value problems and their application to the one-dimensional Schrödinger equation. Applied Mathematics Letters 6 (1993) 67-73
106. Simos T.E. A high order predictor-corrector method for periodic IVPs. Applied Mathematics Letters 6 (1993) 9-12
107. Simos T.E. A modified Runge-Kutta method for the numerical solution of ODEs with oscillation solutions. Applied Mathematics Letters 9 (1996) 61-66
108. Simos T.E. Exponentially-fitted Runge-Kutta-Nyström method for the numerical solution of initial-value problems with oscillating solutions. Applied Mathematics Letters 15 (2002) 217-225
109. Simos T.E. Some new four-step exponential fitting methods for the numerical solution of the radial Schrödinger equation. IMA Journal of Numerical Analysis 11 (1991) 347-356
110. Simos T.E. A fourth algebraic order exponentially-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation. IMA Journal of Numerical Analysis 21 (2001) 919-931
111. Simos T.E. An explicit high order predictor-corrector method for periodic initial-value problems. Mathematical Models and Methods in Applied Sciences 5 (1995) 59-166
112. Simos T.E., Famelis I.T., and Tsitouras C. Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions. Numerical Algorithms 34 1 (2003) 27-40
113. Henrici P. Discrete Variable Methods in Ordinary Differential Equations (1962), John Wiley and Sons, NewYork, USA
114. Ixaru L.Gr., and Micu M. Topics in Theoretical Physics (1978), Central Institute of Physics, Bucharest
115. Landau L.D., and Lifshitz F.M. Quantum Mechanics (1965), Pergamon, New York
116. In: Prigogine I., and Rice S. (Eds). Advances in chemical physics. New Methods in Computational Quantum Mechanics vol. 93 (1997), John Wiley & Sons
117. G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand, Toronto, 1950.
118. Simos T.E. Closed Newton-Cotes trigonometrically-fitted formulae for numerical integration of the Schrödinger equation. Computing Letters 3 1 (2007) 45-57
119. Kalogiratou Z., and Simos T.E. Newton-Cotes formulae for long-time integration. Journal of Computational and Applied Mathematics 158 1 (2003) 75-82
120. Simos T.E. Closed Newton-Cotes trigonometrically-fitted formulae for long-time integration of orbital problems. Revista Mexicana de Astronomia y Astrofisica 42 2 (2006) 167-177
121. Simos T.E. Closed Newton-Cotes trigonometrically-fitted formulae for long-time integration. International Journal of Modern Physics C 14 8 (2003) 1061-1074
122. Simos T.E. High-order closed Newton-Cotes trigonometrically-fitted formulae for long-time integration of orbital problems. Computer Physics Communications 178 3 (2008) 199-207