Numerical multistep methods for the efficient solution of quantum mechanics and related problems

Physics Reports

Volumn 482-483, Issue , 2009, Pages 1-240

  Anastassi, Z A ^a   Simos, T E ^b  

^a TECHNOLOGICAL EDUCATIONAL INSTITUTE OF KALAMATA   (Greece)

^b UNIVERSITY OF PELOPONNESE   (Greece)

Author keywords

0.260; 95.10.E; Energy; Explicit; Exponential fitting; Exponential order; Hybrid; Implicit; Initial value problems; Interval of periodicity; Linear; Local truncation error; Multistep; N body problem; Orbital problems; Ordinary differential equations; Oscillatory; Phase fitting; Phase lag; Predictor corrector; Region of periodicity; Schr dinger equation; Stability analysis; Symmetric; Trigonometric fitting

Indexed keywords

EID: 79952968790     PISSN: 03701573     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physrep.2009.07.005     Document Type: Review

References (152)

1. G.D. Quinlan, Resonances and instabilities in symmetric multistep methods, arXiv: astro-ph/9901136v11999.
2. Quinlan, G.D., Tremaine, S., Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J., 100(5), 1990.
3. Raptis, D., Allison, A.C., Exponential-fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Comm., 14(1), 1978.
4. Raptis, A.D., Exponentially-fitted solutions of the eigenvalue Schrödinger equation with automatic error control. Comput. Phys. Comm., 28(427), 1983.
5. Simos, T.E., A P-stable exponentially-fitted method for the numerical integration of the Schrödinger equation. Molecular Simulation 31:14-15 (2005), 1095–1100.
6. Simos, T.E., A four-step exponentially fitted method for the numerical solution of the Schrödinger equation. J. Math. Chem. 40:3 (2006), 305–318.
7. Kalogiratou, Z., Simos, T.E., A P-stable exponentially-fitted method for the numerical integration of the Schrödinger equation. Appl. Math. Comput. 112 (2000), 99–112.
8. Simos, T.E., Four-step P-stable method with minimal phase-lag. Comput. Phys. Comm., 115(1), 1998.
9. ″ =f(x,y). J. Comput. Appl. Math. 209:1 (2007), 33–53.
10. Van de Vyver, H., An explicit Numerov-type method for second-order differential equations with oscillating solutions. Comput. Math. Appl. 53:9 (2007), 1339–1348.
11. Van de Vyver, H., An adapted explicit hybrid method of Numerov-type for the numerical integration of perturbed oscillators. Appl. Math. Comput. 186:2 (2007), 1385–1394.
12. Vanden Berghe, G., Van Daele, M., Exponentially-fitted Numerov methods. J. Comput. Appl. Math. 200 (2007), 140–153.
13. Simos, T.E., Exponentially-fitted and trigonometrically-fitted symmetric linear multistep methods for the numerical integration of orbital problems. Phys. Lett. A 315 (2003), 437–446.
14. Simos, T.E., An accurate eighth order exponentially-fitted method for the efficient solution of the Schrödinger equation. Comput. Phys. Comm. 125 (2000), 21–59.
15. Simos, T.E., Williams, P.S., A family of Numerov-type exponentially fitted methods for the numerical integration of the Schrödinger equation. Comput. Chem. 21:6 (1997), 403–417.
16. Konguetsof, A., Simos, T.E., An exponentially fitted and trigonometrically fitted method for the numerical solution of periodic initial value problems. Comput. Math. Appl. 45 (2003), 547–554.
17. Simos, T.E., A trigonometrically-fitted method for long-time integration of orbital problems. Math. Comput. Modelling 40 (2004), 1263–1272.
18. Simos, T.E., Vigo-Aguiar, Jesus, An exponentially-fitted high order method for long-term integration of periodic initial-value problems. Comput. Phys. Comm. 140 (2001), 358–365.
19. Simos, T.E., Exponentially-fitted and trigonometrically-fitted methods for long-term integration of orbital problems. New Astron. 7 (2002), 1–7.
20. Tsitouras, Ch., Simos, T.E., Explicit high order methods for the numerical integration of periodic initial-value problems. Appl. Math. Comput. 95 (1998), 15 26–15 26.
21. Simos, T.E., A family of four-step trigonometrically-fitted methods and its application to the Schrödinger equation. J. Math. Chem. 44:2 (2008), 447–466.
22. Simos, T.E., P-stable exponentially fitted methods for the numerical integration of the Schrödinger equation. J. Comput. Phys. 148 (1999), 305–321.
23. P.S. Williams, T.E. Simos, Two-step p-stable methods with phase-lag of order infinity for the numerical solution of special second order initial value problems, 1196, 565–572 (2006).
24. Simos, T.E., Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation. Comput. Chem. 23 (1999), 439–446.
25. Simos, T.E., Trigonometrically-fitted partitioned multistep methods for the integration of orbital problems. New Astron. 9 (2004), 409–415.
26. Simos, T.E., A family of four-step exponentially fitted predictor–corrector methods for the numerical integration of the Schrödinger equation. J. Comput. Appl. Math. 58 (1995), 337–344.
27. Fang, Yonglei, Wu, Xinyuan, A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions. Appl. Numer. Math. 58:3 (2008), 341–351.
28. Psihoyios, G., Simos, T.E., Trigonometrically-fitted symmetric multistep methods for the approximate solution of orbital problems. New Astron. 8 (2003), 679–690.
29. Simos, T.E., Exponentially-fitted and trigonometrically-fitted methods for the numerical solution of orbital problems. New Astron. 8 (2003), 391–400.
30. Psihoyios, G., Simos, T.E., Efficient Numerical Solution of Orbital Problems with the use of Symmetric Four-step Trigonometrically-fitted Methods. Appl. Numer. Anal. Comput. Math. 1:1 (2004), 216–222.
31. Psihoyios, G., Simos, T.E., A new trigonometrically-fitted sixth algebraic order p-c algorithm for the numerical solution of radial Schrödinger equation. Math. Comput. Modelling 42 (2005), 887–902.
32. Thomas, R.M., Simos b, T.E., Mitsou, G.V., A family of Numerov-type exponentially fitted predictor–corrector methods for the numerical integration of the radial Schrödinger equation. J. Comput. Appl. Math. 67 (1996), 255–270.
33. Psihoyios, G., Simos, T.E., Effective numerical approximation of Schrödinger type equations through multiderivative exponentially-fitted schemes. Appl. Numer. Anal. Comput. Math. 1:1 (2004), 205–215.
34. Simos, T.E., Explicit exponentially fitted methods for the numerical solution of the Schrödinger equation. Appl. Math. Comput. 98 (1999), 185–198.
35. Simos, T.E., Raptis, A.D., Nnmerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrödinger equation. Computing 45 (1990), 175–181.
36. 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 (2001), 77–82.
37. Papakaliatakis, G., Simos, T.E., A new finite-difference method with minimal phase lag for the numerical solution of differential equations with engineering applications. Adv. Eng. Soft. 30 (1999), 103–107.
38. Simos, T.E., A new finite difference scheme with minimal phase-lag for the numerical solution of the Schrödinger equation. Appl. Math. Comput. 106 (1999), 245–264.
39. Simos, TE., Eighth-order methods for elastic scattering phase shifts. Internat. J. Theoret. Phys., 36(3), 1997.
40. Simos, T.E., Famelis, I.T., Tsitouras, C., Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions. Numer. Algorithms 34 (2003), 27–40.
41. Simos, T.E., Williams, P.S., A P-stable hybrid exponentially-fitted method for the numerical integration of the Schrödinger equation. Comput. Phys. Comm. 131 (2000), 109–119.
42. Avdelas, G., Kefalidis, E., Simos, T.E., New P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation. Comput. Mater. Sci. 21 (2001), 301–319.
43. Simos, T.E., A family of P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 25:65 (1999), 65–84.
44. Aguiar, J.V., Simos, T.E., A family of P-stable eighth algebraic order methods with exponential fitting facilities. J. Math. Chem., 29(3), 2001.
45. Simos, T.E., Dissipative trigonometrically fitted methods for the numerical solution of orbital problems. New Astron. 9 (2003), 59–68.
46. Chawla, M.M., Rao, P.S., A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II. Explicit method. J. Comput. Appl. Math., 15(329), 1986.
47. Henrici, P., Discrete Variable Methods in Ordinary Differential Equations. 1962, John Wiley and Sons, New York, USA.
48. Lyche, T., Chebyshevian multistep methods for Ordinary Differential Equations. Numer. Math. 19 (1972), 65–75.
49. Lambert, J.D., Watson, I.A., Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18 (1976), 189–202.
50. Gr, L, Rizea, Ixaru, M., A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Comput. Phys. Comm. 19 (1980), 23–27.
51. Gr Ixaru, L., Rizea, M., Comparison of some four-step methods for the numerical solution of the Schrödinger equation. Comput. Phys. Comm. 38:3 (1985), 329–337.
52. Simos, T.E., A 4-step method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 30:3 (1990), 251–255.
53. Papageorgiou, C.D., Raptis, A.D., Simos, T.E., A method for computing phase-shifts for scattering. J. Computat. Appl. Math. 29:1 (1990), 61–67.
54. Raptis, A.D., Simos, T.E., A 4-step phase-fitted method for the numerical-integration of 2nd-order initial-value problems. BIT 31:1 (1991), 160–168.
55. Simos, T.E., Some new 4-step exponential-fitting methods for the numerical-solution of the radial Schrödinger equation. IMA J. Numer. Anal. 11:3 (1991), 347–356.
56. Simos, T.E., A numerov-type method for the numerical-solution of the radial Schrd¨inger-equation. Appl. Numer. Math. 7:2 (1991), 201–206.
57. De Meyer, H., Vanden Berghe, G., Vanthournout, J., Modified backward differentiation methods of the adams-type based on exponential interpolation. Computers Math. Applic. 21:2-3 (1991), 171–179.
58. Sideridis, AB, Simos, T.E., A low-order embedded Runge–Kutta method for periodic initial-value problems. J. Comput. Appl. Math. 44:2 (1992), 235–244.
59. Simos, T.E., Raptis, AD, A 4th-order bessel fitting method for the numerical-solution of the Schrd¨inger-equation. J. Computat. Appl. Math. 43:3 (1992), 313–322.
60. Simos, T.E., Explicit 2-step methods with minimal phase-lag for the numerical-integration of special 2nd-order initial-value problems and their application to the one-dimensional Schrd¨inger-equation. J. Comput. Appl. Math. 39:1 (1992), 89–94.
61. Simos, T.E., Runge–Kutta–Nyström interpolants for the numerical integration of special 2nd-order periodic initial-value problems. Comput. Math. Appl. 26:12 (1993), 7–15.
62. Simos, T.E., Runge–Kutta interpolants with minimal phase-lag. Comput. Math. Appl. 26:8 (1993), 43–49.
63. Simos, T.E., A high-order predictor–corrector method for periodic IVPS. Appl. Math. Lett. 6:5 (1993), 9–12.
64. 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 (1993), 67–73.
65. 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 25:6 (1993), 95–101.
66. Simos, T.E., An explicit 4-step phase-fitted method for the numerical-integration of 2nd-order initial-value problems. J. Computat. Appl. Math. 55:2 (1994), 125–133.
67. Simos, T.E., Dimas, E, Sideridis, Ab, A Runge–Kutta–Nystrom method for the numerical-integration of special 2nd-order periodic initial-value problems. J. Comput. Appl. Math. 51:3 (1994), 317–326.
68. Simos, T.E., Mitsou, G.V., A family of 4-step exponential fitted methods for the numerical-integration of the radial Schrödinger-equation. Computers and Mathematics With Applications 28:4 (1994), 41–50.
69. ″ =F(X,Y) and an application to the Schrödinger equation. Int. J. Quant. Chem. 53 (1995), 473–483.
70. Simos, T.E., Mousadis, G., A 2-step method for the numerical-solution of the radial Schrödinger-equation. Computers and Mathematics With Applications 29:7 (1995), 31–37.
71. Simos, T.E., An explicit high-order predictor–corrector method for periodic initial-value problems. Math. Models and Methods Appl. Sci. 5:2 (1995), 159–166.
72. Avdelas, G., Simos, T.E., A generator of high-order embedded P-stable methods for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 72:2 (1996), 345–358.
73. Avdelas, G., Simos, T.E., Embedded methods for the numerical solution of the Schrödinger equation. Computers and Mathematics With Applications 31:2 (1996), 85–102.
74. Avdelas, G., Simos, T.E., Block Runge–Kutta methods for periodic initial-value problems. Computers and Mathematics With Applications 31:1 (1996), 69–83.
75. Papakaliatakis, G., Simos, T.E., A new method for the numerical solution of fourth-order BVP's with oscillating solutions. Comput. Math. Appl. 32:10 (1996), 1–6.
76. ″ =f(x,y). IMA J. Numer. Anal. 16 (1996), 179–199.
77. Simos, T.E., A modified Runge–Kutta method for the numerical solution of ODEs with oscillation solutions. Appl. Math. Lett. 9:6 (1996), 61–66.
78. Simos, T.E., Williams, P.S., A finite-difference method for the numerical solution of the Schrödinger equation. J. Computat. Appl. Math. 79:2 (1997), 189–205.
79. Thomas, R.M., Simos, T.E., A family of hybrid exponentially fitted predictor–corrector methods for the numerical integration of the radial Schrödinger equation. J. Comput. Appl. Math. 87 (1997), 215–226.
80. Simos, T.E., Tsitouras, Ch., A P-Stable eighth order method for the numerical integration of periodic initial value problems. J. Comput. Phys. 130 (1997), 123–128.
81. Simos, T.E., An extended Numerov-type method for the numerical solution of the Schrödinger equation. Computers and Mathematics With Applications 33:10 (1997), 67–78.
82. Simos, T.E., A new hybrid imbedded variable-step procedure for the numerical integration of the Schrödinger equation. Computers and Mathematics With Applications 36:2 (1998), 51–63.
83. Simos, T.E., An accurate finite difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 91:1 (1998), 47–61.
84. Simos, T.E., An exponentially fitted eighth-order method for the numerical solution of the Schrödinger equation. J. Computat. Appl. Math. 108:1-2 (1999), 177–194.
85. Simos, T.E., Chemical Modelling — Applications and Theory Vol.1, Specialist Periodical Reports. 2000, The Royal Society of Chemistry, Cambridge.
86. Simos, T.E., A fourth algebraic order exponentially-fitted Runge–Kutta method for the numerical solution of the Schrödinger equation. IMA J. Numer. Anal. 21:4 (2001), 919–931.
87. Simos, T.E., Bessel and Neumann fitted methods for the numerical solution of the Schrödinger equation. Comput. Math. Appl. 42:6-7 (2001), 833–847.
88. Simos, T.E., Williams, P.S., Dissipative exponentially-fitted methods for the numerical solution of the Schrödinger equation. Comput. Chem. 25 (2001), 261–273.
89. Tsitouras, C., Simos, T.E., Optimized Runge–Kutta pairs for problems with oscillating solutions. J. Computat. Appl. Math. 147:2 (2002), 397–409.
90. Simos, T.E., Vigo-Aguiar, Jesus, Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrödinger equation. J. Math. Chem., 31(2), 2002.
91. 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 (2002), 123–137.
92. Simos, T.E., Exponentially-fitted Runge–Kutta-Nystrom method for the numerical solution of initial-value problems with oscillating solutions. Appl. Math. Lett. 15:2 (2002), 217–225.
93. Hairer, E., Lubich, C., Wanner, G., Geometric numerical integration, Structure preserving algorithms for ordinary differential equations. 2002, Springer.
94. Simos, T.E., Vigo-Aguiar, J., 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), 2003 (special Issue).
95. Simos, T.E., Avdelas, G., Vigo-Aguiar, J., Numerical methods in physics, chemistry, and engineering — Preface. Computers and Mathematics With Applications, 45(1–3), 2003.
96. Butcher, J.C., Numerical Methods for Ordinary Differential Equations. 2003, Wiley.
97. Williams, P.S., Simos, T.E., A new family of exponentially fitted methods. Math. Comput. Modelling 38 (2003), 571–584.
98. Konguetsof, A., Simos, T.E., A generator of hybrid symmetric four-step methods for the numerical solution of the Schrödinger equation. J. Computat. Appl. Math. 158:1 (2003), 93–106.
99. Gr Ixaru, L., Vanden Berghe, G., De Meyer, H., Exponentially fitted variable two-step BDF algorithm for first order ODEs. Comput. Phys. Comm. 150 (2003), 116–128.
100. Psihoyios, G., Simos, T.E., Trigonometrically fitted predictor–corrector methods for IVPs with oscillating solutions. J. Comput. Appl. Math. 158 (2003), 135–144.
101. Simos, T.E., Vigo-Aguiar, J., A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems. Comput. Phys. Comm. 152 (2003), 274–294.
102. Kalogiratou, Z., Monovasilis, T., Simos, T.E., Symplectic integrators for the numerical solution of the Schrödinger equation. J. Computat. Appl. Math. 158:1 (2003), 83–92.
103. Kalogiratou, Z., Simos, T.E., Newton–Cotes formulae for long-time integration. J. Computat. Appl. Math. 158:1 (2003), 75–82.
104. Simos, T.E., Multiderivative methods for the numerical solution of the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 50 (2004), 7–26.
105. Simos, T.E., Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution. Appl. Math. Lett. 17 (2004), 601–607.
106. Enright, W.H., On the use of ‘arc length’ and ‘defect’ for mesh selection for differential equations. Computing Letters 1:2 (2005), 47–52.
107. Psihoyios, G., Simos, T.E., A family of fifth algebraic order trigonometrically fitted P-C schemes for the numerical solution of the radial Schrödinger equation. MATCH Commun. Math. Comput. Chem. 53 (2005), 321–344.
108. Sakas, D.P., Simos, T.E., Trigonometrically-fitted multiderivative methods for the numerical solution of the radial Schrödinger equation. MATCH Commun. Math. Comput. Chem. 53 (2005), 299–320.
109. Sakas, DP, 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 (2005), 161–172.
110. Zhongcheng, Wang, P-stable linear symmetric multistep methods for periodic initial-value problems. Comput. Phys. Comm. 171 (2005), 162–174.
111. Psihoyios, G., Simos, T.E., A fourth algebraic order trigonometrically fitted predictor–corrector scheme for IVPs with oscillating solutions. J. Comput. Appl. Math. 175 (2005), 137–147.
112. Simos, T.E., Psihoyios, G., 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), 2005 (special issue).
113. Simos, T.E., Gutman, I., Papers presented on the International Conference on Computational Methods in Sciences and Engineering (Castoria, Greece, September 12–16, 2003). MATCH Commun. Math. Comput. Chem., 53(2), 2005, A3–A4.
114. Simos, T.E., P-stable four-step exponentially-fitted method for the numerical integration of the Schrödinger equation. CoLe 1:1 (2005), 37–45.
115. Anastassi, Z.A., Simos, T.E., Trigonometrically Fitted Fifth Order Runge–Kutta Methods for the Numerical Solution of the Schrödinger Equation. Math. Comput. Modelling 42:7–8 (2005), 877–886.
116. Anastassi, Z.A., Simos, T.E., Trigonometrically-fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 3 (2005), 281–293.
117. Anastassi, ZA, Simos, T.E., An optimized Runge–Kutta method for the solution of orbital problems. J. Comput. Appl. Math., 175(1), 2005.
118. Tselios, K., Simos, T.E., Runge–Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. J. Computat. Appl. Math. 175:1 (2005), 173–181.
119. Shampine, L.F., Muir, P.H., Xu, H., A user-friendly fortran BVP solver. J. Numer. Anal. Indust. Appl. Math. 1:2 (2006), 201–217.
120. Psihoyios, G., A block implicit advanced step-point (BIAS) algorithm for stiff differential systems. CoLe 2:1–2 (2006), 51–58.
121. Amodio, P., Gladwell, I., Romanazzi, G., Numerical solution of general bordered ABD linear systems by cyclic reduction. J. Numer. Anal. Indust. Appl. Math. 1:1 (2006), 5–12.
122. Capper, S.D., Cash, J.R., Moore, D.R., Lobatto–Obrechkoff formulae for 2nd order two-point boundary value problems. J. Numer. Anal. Indust. Appl. Math. 1:1 (2006), 13–25.
123. Capper, S.D., Moore, D.R., On high order MIRK schemes and Hermite–Birkhoff interpolants. J. Numer. Anal. Indust. Appl. Math. 1:1 (2006), 27–47.
124. Cash, J.R., Sumarti, N., Abdulla, T.J., Vieira, I., The derivation of interpolants for nonlinear two-point boundary value problems. J. Numer. Anal. Indust. Appl. Math. 1:1 (2006), 49–58.
125. Cash, J.R., Girdlestone, S., Variable step Runge–Kutta–Nyström methods for the numerical solution of reversible systems. J. Numer. Anal. Indust. Appl. Math. 1:1 (2006), 59–80.
126. Cash, J.R., Mazzia, F., Hybrid mesh selection algorithms based on conditioning for two-point boundary value problems. J. Numer. Anal. Indust. Appl. Math. 1:1 (2006), 81–90.
127. Iavernaro, F., Mazzia, F., Trigiante, D., Stability and conditioning in numerical analysis. J. Numer. Anal. Indust. Appl. Math. 1:1 (2006), 91–112.
128. Iavernaro, F., Trigiante, D., Discrete conservative vector fields induced by the trapezoidal method. J. Numer. Anal. Indust. Appl. Math. 1:1 (2006), 113–130.
129. Mazzia, F., Sestini, A., Trigiante, D., BS linear multistep methods on non-uniform meshes. J. Numer. Anal. Indust. Appl. Math. 1:1 (2006), 131–144.
130. Vanden Berghe, G., Van Daele, M., Exponentially-fitted stormer/verlet methods. J. Numer. Anal. Indust. Appl. Math. 1:3 (2006), 241–255.
131. Simos, T.E., Psihoyios, G., The International Conference on Computational Methods in Sciences and Engineering 2004 — Preface. J. Comput. Appl. Math., 191(2), 2006, 165 (special issue).
132. Aceto, L., Pandolfi, R., Trigiante, D., Stability analysis of linear multistep methods via polynomial type variation. J. Numer. Anal. Indust. Appl. Math. 2:1-2 (2007), 1–9.
133. Simos, T.E., Closed Newton–Cotes trigonometrically-fitted formulae for numerical integration of the Schrödinger equation. CoLe 1:3 (2007), 45–57.
134. Simos, T.E., High-order closed Newton–Cotes trigonometrically-fitted formulae for long-time integration of orbital problems. Comput. Phys. Comm. 178 (2008), 199–207.
135. Anastassi, Z.A., Simos, T.E., New trigonometrically fitted six-step symmetric methods for the efficient solution of the Schrödinger equation. MATCH Commun. Math. Comput. Chem., 60(3), 2008.
136. 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), 2008.
137. Anastassi, Z.A., Simos, T.E., A six-step p-stable trigonometrically-fitted method for the numerical integration of the radial Schrödinger equation. MATCH Commun. Math. Comput. Chem., 60(3), 2008.
138. Corless, Robert M., Shakoori, Azar, Aruliah, D.A., Laureano Gonzalez-Vega, Barycentric hermite interpolants for event location in initial-value problems. J. Numer. Anal. Indust. Appl. Math. 3(:1-2 (2008), 1–16.
139. Dewar, Mike, Embedding a general-purpose numerical library in an interactive environment. J. Numer. Anal. Indust. Appl. Math. 3:1–2 (2008), 17–26.
140. Kierzenka, J., Shampine, L.F., A BVP solver that controls residual and error. J. Numer. Anal. Indust. Appl. Math. 3:1–2 (2008), 27–41.
141. Knapp, R., A method of lines framework in Mathematica. J. Numer. Anal. Indust. Appl. Math. 3:1–2 (2008), 43–59.
142. Nedialkov, N.S., Pryce, J.D., Solving differential algebraic equations by Taylor series (iii): The code. J. Numer. Anal. Indust. Appl. Math. 3:1–2 (2008), 61–80.
143. Lipsman, Ronald L., Osborn, John E., Rosenberg, Jonathan M., The SCHOL project at the university of maryland: Using mathematical software in the teaching of sophomore Differential Equations. J. Numer. Anal. Indust. Appl. Math. 3:1–2 (2008), 81–103.
144. Sofroniou, Mark, Spaletta, Giulia, Extrapolation methods in Mathematica. J. Numer. Anal. Indust. Appl. Math. 3:1–2 (2008), 105–121.
145. Spiteri, Raymond J., Ter, Thian-Peng, A PSE for the numerical solution of nonlinear algebraic equations. J. Numer. Anal. Indust. Appl. Math. 3:1–2 (2008), 123–137.
146. Corwin, S.P., Thompson, S., White, S.M., Solving ODEs and DDEs with impulses. J. Numer. Anal. Indust. Appl. Math. 3:1–2 (2008), 139–150.
147. Weckesser, Warren, VFGEN: A code generation tool. J. Numer. Anal. Indust. Appl. Math. 3:1–2 (2008), 151–164.
148. Wittkopf, Allan, Automatic code generation and optimization in maple. J. Numer. Anal. Indust. Appl. Math. 3:1–2 (2008), 165–178.
149. http://www.burtleburtle.net/bob/math/multistep.html.
150. Anastassi, Z.A., 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. J. Math. Chem. 41:1 (2007), 79–100.
151. Anastassi, Z.A., Simos, T.E., A family of two-stage two-step methods for the numerical integration of the Schrödinger equation and related IVPs with oscillating solution. J. Math. Chem. 45:4 (2008), 1102–1129.
152. Z.A. Anastassi, D.S. Vlachos, T.E. Simos, A family of multistep methods with zero phase-lag and derivatives for the numerical integration of oscillatory ODEs, J. Math. Chem. (in press).