Phase-fitted and amplification-fitted two-step hybrid methods for y″ = f (x, y)

Journal of Computational and Applied Mathematics

Volumn 209, Issue 1, 2007, Pages 33-53

  Van de Vyver, Hans ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

Oscillating solutions; Phase fitting; Second order initial value problems; Two step hybrid methods; Variable coefficients

Indexed keywords

INITIAL VALUE PROBLEMS; NUMERICAL METHODS; PARAMETER ESTIMATION;

OSCILLATING SOLUTIONS; PHASE FITTING; TWO-STEP HYBRID METHODS; VARIABLE COEFFICIENTS;

FUNCTION EVALUATION;

EID: 34548046245     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2006.10.025     Document Type: Article

References (35)

1. Anastassi Z.A., and Simos T.E. Special optimized Runge-Kutta methods for IVP's with oscillating solutions. Internat. J. Modern Phys. C 15 (2004) 1-15
2. Anastassi Z.A., and Simos T.E. A dispersive-fitted and dissipative-fitted explicit Runge-Kutta method for the numerical solution of orbital problems. New Astronom. 10 (2004) 31-37
3. Avdelas G., and Simos T.E. Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation. Phys. Rev. E 62 (2000) 1375-1381
4. A.G. Bratsos, I. Th. Famelis, A. Kollias, Ch. Tsitouras, Phase-fitted Numerov type methods, Appl. Math. Comput., in press.
5. Brusa L., and Nigro L. A one-step method for direct integration of structural dynamic equations. Internat. J. Numer. Methods Engrg. 15 (1980) 685-699
6. ″ = f (x, y). Appl. Numer. Math. 53 (2005) 201-212
7. Chawla M.M., and Rao P.S. A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. J. Comput. Appl. Math. 11 (1984) 277-281
8. ″ = f (t, y). J. Comput. Appl. Math. 17 (1987) 365-368
9. ″ = f (x, y). BIT 21 (1981) 455-464
10. ″ = f (x, y). IMA J. Numer. Anal. 23 (2003) 197-220
11. ″ = f (x, y). J. Comput. Appl. Math. 126 (2000) 47-75
12. ″ = f (x, y). IMA J. Numer. Anal. 16 (1996) 179-199
13. Franco J.M. A class of explicit two-step hybrid methods for second-order IVPs. J. Comput. Appl. Math. 187 (2006) 41-57
14. Franco J.M., and Palacios M. High-order P-stable multistep methods. J. Comput. Appl. Math. 30 (1990) 1-10
15. Ixaru L.Gr., and Rizea M. Numerov method maximally adapted to the Schrödinger equation. J. Comput. Phys. 73 (1987) 306-324
16. Ixaru L.Gr., and Vanden Berghe G. Exponential Fitting (2004), Kluwer Academic Publishers, Dordrecht, Boston, London
17. Lambert J.D. Computational Methods in Ordinary Differential Equations (1973), Wiley, London
18. Lambert J.D., and Watson I.A. Symmetric multistep methods for periodic initial value problems. J. Inst. Math. Appl. 18 (1976) 189-202
19. Ozawa K. A four-stage implicit Runge-Kutta-Nyström method with variable coefficients for solving periodic initial value problems. Jpn. J. Indust. Appl. Math. 16 (1999) 25-46
20. Papageorgiou G., Tsitouras Ch., and Famelis I. Explicit Numerov type methods for second order IVPs with oscillating solutions. Internat. J. Modern Phys. C 12 (2001) 657-666
21. Paternoster B. A phase-fitted collocation based Runge-Kutta-Nyström method. Appl. Numer. Math. 35 (2000) 339-355
22. 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
23. Simos T.E. A two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problems. Internat. J. Comput. Math. 39 (1991) 135-140
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. Tsitouras Ch. Explicit two-step methods for second-order linear IVPs. Comput. Math. Appl. 43 (2002) 943-949
26. Tsitouras Ch. Explicit Numerov type methods with reduced number of stages. Comput. Math. Appl. 45 (2003) 37-42
27. Van de Vyver H. Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems. Comput. Phys. Comm. 173 (2005) 115-130
28. Van de Vyver H. Frequency evaluation for exponentially-fitted Runge-Kutta methods. J. Comput. Appl. Math. 184 (2005) 442-463
29. Van de Vyver H. An embedded phase-fitted modified Runge-Kutta method for the numerical solution of the Schrödinger equation. Phys. Lett. A 352 (2006) 278-285
30. Van de Vyver H. On the generation of P-stable exponentially fitted Runge-Kutta-Nyström methods by exponentially fitted Runge-Kutta methods. J. Comput. Appl. Math. 188 (2006) 309-318
31. Van de Vyver H. A fourth-order symplectic exponentially fitted integrator. Comput. Phys. Comm. 174 (2006) 255-262
32. Van de Vyver H. A phase-fitted and amplification-fitted explicit two-step hybrid method for second-order periodic initial value problems. Internat. J. Modern Phys. C 17 (2006) 663-675
33. van der Houwen P.J., and Sommeijer B.P. Explicit Runge-Kutta(-Nyström) methods with reduced phase errors for computing oscillating solutions. SIAM J. Numer. Anal. 24 (1987) 595-617
34. Van Dooren R. Stabilization of Cowell's classical finite difference methods for numerical integration. J. Comput. Phys. 16 (1974) 186-192
35. Vigo-Aguiar J., Simos T.E., and Ferrándiz J.M. Controlling the error growth in long-term numerical integration of perturbed oscillations in one or more frequencies. Proc. Roy. Soc. London Ser. A 460 (2004) 561-567