Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation

Acta Applicandae Mathematicae

Volumn 110, Issue 3, 2010, Pages 1331-1352

  Simos, T E ^a,b  

^a Laboratory of Computational Sciences Faculty of Sciences and Technology University of Peloponnese ^*  (Greece)

^b UNIVERSITY OF PELOPONNESE   (Greece)

Author keywords

Exponential fitting; Hybrid methods; Multistep methods; Numerical solution; Schr dinger equation; Trigonometric fitting

Indexed keywords

ORDINARY DIFFERENTIAL EQUATIONS;

DINGER EQUATION; EXPONENTIAL FITTING; HYBRID METHOD; MULTI STEP METHODS; NUMERICAL SOLUTION; TRIGONOMETRIC FITTING;

EXPONENTIAL FUNCTIONS;

EID: 77957331657     PISSN: 01678019     EISSN: 15729036     Source Type: Journal    
DOI: 10.1007/s10440-009-9513-6     Document Type: Article

References (65)

1. Aceto, L., Pandolfi, R., Trigiante, D.: Stability analysis of linear multistep methods via polynomial type variation. JNAIAM J. Numer. Anal. Ind. Appl. Math. 2(1-2), 1-9 (2007)
2. Anastassi, Z.A., Simos, T.E.: Trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 281-293 (2005)
3. 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), 79-100 (2007)
4. 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)
5. 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)
6. 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)
7. Avdelas, G., Kefalidis, E., Simos, T.E.: New P-stable eighth algebraic order exponentially-fitted methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 31(4), 371-404 (2002)
8. Chawla, M.M.: Unconditionally stable Noumerov-type methods for second order differential equations. BIT 23, 541-542 (1983)
9. Corless, R.M., Shakoori, A., Aruliah, D.A., Gonzalez-Vega, L.: Barycentric Hermite interpolants for event location in initial-value problems. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 1-16 (2008)
10. Corwin, S.P., Thompson, S., White, S.M.: Solving ODEs and DDEs with impulses. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 139-149 (2008)
11. Dewar, M.: Embedding a general-purpose numerical library in an interactive environment. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 17-26 (2008)
12. Enright, W.H.: On the use of 'arc length' and 'defect' for mesh selection for differential equations. Comput. Lett. 1(2), 47-52 (2005)
13. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York (1962)
14. Ixaru, L.Gr.: Numerical Methods for Differential Equations and Applications. Reidel, Dordrecht (1984)
15. 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)
16. 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)
17. Ixaru, L.Gr., Vanden Berghe, G.: Exponential Fitting. Series on Mathematics and Its Applications, vol. 568. Kluwer Academic, Norwell (2004)
18. Kalogiratou, Z., Simos, T.E.: A P-stable exponentially-fitted method for the numerical integration of the Schrödinger equation. Appl. Math. Comput. 112, 99-112 (2000)
19. Kalogiratou, Z., Simos, T.E.: Construction of trigonometrically and exponentially fitted Runge-Kutta-Nyström methods for the numerical solution of the Schrödinger equation and related problems a method of 8th algebraic order. J. Math. Chem. 31(2), 211-232
20. Kalogiratou, Z., Monovasilis, T., Simos, T.E.: Numerical solution of the two-dimensional time independent Schrödinger equation with Numerov-type methods. J. Math. Chem. 37(3), 271-279 (2005)
21. Kierzenka, J., Shampine, L.F.: A BVP solver that controls residual and error. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 27-41 (2008)
22. Knapp, R.: A method of lines framework in Mathematica. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 43-59 (2008)
23. Konguetsof, A., Simos, T.E.: On the construction of exponentially-fitted methods for the numerical solution of the Schrödinger equation. J. Comput. Methods Sci. Eng. 1, 143-165 (2001)
24. Lambert, J.D., Watson, I.A.: Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189-202 (1976)
25. Lipsman, R.L., Osborn, J.E., Rosenberg, J.M.: The SCHOL project at the University of Maryland: using mathematical software in the teaching of sophomore differential equations. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 81-103 (2008)
26. Monovasilis, T., Kalogiratou, Z., Simos, T.E.: Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 37(3), 263-270 (2005)
27. Monovasilis, T., Kalogiratou, Z., Simos, T.E.: Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 40(3), 257-267 (2006)
28. Nedialkov, N.S., Pryce, J.D.: Solving differential algebraic equations by Taylor series (III): the DAETS code. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 61-80 (2008)
29. Psihoyios, G.: A Block Implicit Advanced Step-point (BIAS) algorithm for stiff differential systems. Comput. Lett. 2(1-2), 51-58 (2006)
30. Psihoyios, G., Simos, T.E.: Sixth algebraic order trigonometrically fitted predictor-corrector methods for the numerical solution of the radial Schrödinger equation. J. Math. Chem. 37(3), 295-316 (2005)
31. Psihoyios, G., Simos, T.E.: The numerical solution of the radial Schrödinger equation via a trigonometrically fitted family of seventh algebraic order Predictor-Corrector methods. J. Math. Chem. 40(3), 269-293 (2006)
32. Raptis, A.D.: On the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 24, 1-4 (1981)
33. Raptis, A.D.: Exponentially-fitted solutions of the eigenvalue Shrödinger equation with automatic error control. Comput. Phys. Commun. 28, 427-431 (1983)
34. Raptis, A.D.: Exponential multistep methods for ordinary differential equations. Bull. Greek Math. Soc. 25, 113-126 (1984)
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., Simos, T.E.: A four-step phase-fitted method for the numerical integration of second order initial-value problem. BIT 31, 160-168 (1991)
37. 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)
38. Simos, T.E.: Numerical solution of ordinary differential equations with periodical solution. Doctoral Dissertation, National Technical University of Athens, Greece (1990) (in Greek)
39. 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)
40. 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)
41. Simos, T.E.: A family of P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 25(1), 65-84 (1999)
42. Simos, T.E.: A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation. J. Math. Chem. 27(4), 343-356 (2000)
43. Simos, T.E.: Atomic structure computations. In: Hinchliffe, A. (ed.) Chemical Modelling: Applications and Theory, pp. 38-142. Royal Soc. Chem., London (2000)
44. Simos, T.E.: Numerical methods for 1D, 2D and 3D differential equations arising in chemical problems. In: Chemical Modelling: Application and Theory, vol. 2, pp. 170-270. Royal Soc. Chem., London (2002)
45. Simos, T.E.: A family of trigonometrically-fitted symmetric methods for the efficient solution of the Schrödinger equation and related problems. J. Math. Chem. 34(1-2), 39-58 (2003)
46. Simos, T.E.: Numerical methods in chemistry. In: Chemical Modelling: Application and Theory, vol. 3, pp. 271-378. Royal Soc. Chem., London 2004)
47. Simos, T.E.: Multiderivative methods for the numerical solution of the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 45, 7-26 (2004)
48. Simos, T.E.: Exponentially - fitted multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 36(1), 13-27 (2004)
49. Simos, T.E.: A four-step exponentially fitted method for the numerical solution of the Schrödinger equation. J. Math. Chem. 40(3), 305-318 (2006)
50. Simos, T.E.: Numerical methods in chemistry. In: Chemical Modelling: Application and Theory, vol. 4, pp. 161-244. Royal Soc. Chem., London (2006)
51. Simos, T.E.: Stabilization of a four-step exponentially-fitted method and its application to the Schrödinger equation. Int. J. Mod. Phys. C 18(3), 315-328 (2007)
52. Simos, T.E.: Numerical methods in chemistry. In: Chemical Modelling: Application and Theory, vol. 5, pp. 350-487. Royal Soc. Chem., London (2008)
53. Simos, T.E.: In: Numerical methods in chemistry. Chemical Modelling: Application and Theory, vol. 6. Royal Soc. Chem., London (2009, to appear)
54. 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)
55. 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)
56. Simos, T.E., Williams, P.S.: On finite difference methods for the solution of the Schrödinger equation. Comput. Chem. 23, 513-554 (1999)
57. 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)
58. Sofroniou, M., Spaletta, G.: Extrapolation methods in Mathematica. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 105-121 (2008)
59. Spiteri, R.J., Ter, T.-P.: pythNon: a PSE for the numerical solution of nonlinear algebraic equations. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 123-137 (2008)
60. Tselios, K., Simos, T.E.: Symplectic methods for the numerical solution of the radial Shrödinger equation. J. Math. Chem. 34(1-2), 83-94 (2003)
61. Tselios, K., Simos, T.E.: Symplectic methods of fifth order for the numerical solution of the radial Shrödinger equation. J. Math. Chem. 35(1), 55-63 (2004)
62. Vigo-Aguiar, J., Simos, T.E.: A family of P-stable eighth algebraic order methods with exponential fitting facilities. J. Math. Chem. 29(3), 177-189 (2001)
63. Vigo-Aguiar, J., Simos, T.E.: Family of twelve steps exponential fitting symmetric multistep methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 32(3), 257-270 (2002)
64. Weckesser, W.: VFGEN: A code generation tool. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 151-165 (2008)
65. Wittkopf, A.: Automatic code generation and optimization in Maple. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 167-180 (2008)