Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials

Applied Numerical Mathematics

Volumn 28, Issue 2-4, 1998, Pages 401-412

  Paternoster, B ^a  

^a UNIVERSITY OF SALERNO   (Italy)

Author keywords

Kutta methods; Numerical methods for ODEs; Runge

Indexed keywords

INITIAL VALUE PROBLEMS; INTEGRATION; ORDINARY DIFFERENTIAL EQUATIONS; POLYNOMIALS; PROBLEM SOLVING;

RUNGE KUTTA NYSTROM METHODS; TRIGONOMETRIC POLYNOMIALS;

RUNGE KUTTA METHODS;

EID: 0032183736     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-9274(98)00056-7     Document Type: Article

References (25)

1. P. Albrecht, The extension of the theory of A-methods to RK methods, in: K. Strehmel, ed., Numerical Treatment of Differential Equations, Proc. 4th Seminar NUMDIFF-4, Teubner-Texte zur Mathematik (Teubner, Leipzig, 1987) 8-18.
2. P. Albrecht, A new theoretical approach to RK methods, SIAM J. Numer. Anal. 24 (2) (1987) 391-406.
3. J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (Wiley, Chichester, 1987).
4. J.P. Coleman, Numerical methods for y″ = f (x, y) via rational approximations for the cosine, IMA J. Numer. Anal. 9 (1989) 145-165.
5. J.P. Coleman and L.Gr. Ixaru, P-stability and exponential-fitting methods for y″ = f (x, y), IMA J. Numer. Anal. 16 (1996) 179-199.
6. M.R. Crisci, B. Paternoster and E. Russo, Fully parallel Runge-Kutta Nyström methods for ODEs with oscillating solutions, Appl. Numer. Math. 11 (1993) 143-158.
7. M.R. Crisci and B. Paternoster, Parallel Runge-Kutta Nyström methods, Ricerche Mat., to appear.
8. K. Dekker and J.G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, CWI Monograph (North-Holland, Amsterdam, 1984).
9. W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math. 3 (1961) 381-397.
10. E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics 8 (Springer, Berlin, 1987).
11. L.Gr. Ixaru, Numerical Methods for Differential Equations and Applications (Reidel, Dordrecht, 1984).
12. L.Gr. Ixaru, Operations on oscillatory functions, Comput. Phys. Comm. 105 (1997) 1-19.
13. B. Neta and C.H. Ford, Families of methods for ordinary differential equations based on trigonometric polynomials, J. Comput. Appl. Math. 10 (1984) 33-38.
14. A.D. Raptis and A.C. Allison, Exponential-fitting methods for the numerical solution of the Schrödinger equation, Comput. Phys. Comm. 44 (1978) 95-103.
15. A.D. Raptis and T.E. Simos, A four-step phase-fitted method for the numerical integration of second order initial-value problems, BIT 31 (1991) 160-168.
16. T.E. Simos, E. Dimas and A.B. Sideridis, A Runge-Kutta-Nyström method for the numerical integration of special second-order periodic initial-value problems, J. Comput. Appl. Math. 51 (1994) 317-326.
17. B.P. Sommeijer, A note on a diagonally implicit Runge-Kutta-Nyström method, J. Comput. Appl. Math. 19 (1987) 395-399.
18. E. Stiefel and D.G. Bettis, Stabilization of Cowell's method, Numer. Math. 13 (1969) 154-175.
19. M. van Daele, H. de Meyer and G. vanden Berghe, A modified Numerov integration method for general second order initial value problems, Internat. J. Comput. Math. 40 (1991) 117-127.
20. G. vanden Berghe, H. de Meyer and J. Vanthournout, A modified Numerov integration method for second order periodic initial-value problem, Internat. J. Comput. Math. 32 (1990) 233-242.
21. P.J. van der Houwen and B.P. Sommeijer, Linear multistep methods with reduced truncation error for periodic initial-value problem, IMA J. Numer. Anal. 4 (1984) 479-489.
22. P.J. van der Houwen and B.P. Sommeijer, Explicit Runge-Kutta(-Nyström) methods with reduced phase errors for computing oscillating solution, SIAM J. Numer. Anal. 24 (1987) 595-617.
23. P.J. van der Houwen and B.P. Sommeijer, Phase-lag analysis of implicit Runge-Kutta methods, SIAM J. Numer. Anal. 26 (1989) 214-228.
24. P.J. van der Houwen and B.P. Sommeijer, Diagonally implicit Runge-Kutta Nyström methods for oscillatory problems, SIAM J. Numer. Anal 26 (1989) 414-429.
25. P.J. van der Houwen, B.P. Sommeijer, K. Strehmel and R.Weiner, On the numerical integration of second order initial value problems with a periodic forcing function, Computing 37 (1986) 195-218.