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Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials. (English) Zbl 0927.65097
The paper is concerned with initial value problems for ordinary differential equations (ODEs) whose solutions are a priori known to oscillate with a known frequency. Runge-Kutta methods are applied to first-order problems and Runge-Kutta-Nyström methods to second-order problems. {\it P. Albrecht}’s approach [SIAM J. Numer. Anal. 24, 391-406 (1987; Zbl 0617.65067); Teubner-Texte Math. 104, 8-18 (1988; Zbl 0682.65041)] is considered and the concept of trigonometric order of the method is used. The Runge-Kutta and Runge-Kutta-Nyström methods derived in this paper have low orders (1 or 2) and integrate trigonometric polynomials exactly. Two numerical examples are performed for comparison to others-Runge-Kutta-Nyström methods.

65L06Multistep, Runge-Kutta, and extrapolation methods
65L05Initial value problems for ODE (numerical methods)
34C25Periodic solutions of ODE
34A34Nonlinear ODE and systems, general
Full Text: DOI
[1] Albrecht, P.: The extension of the theory of A-methods to RK methods. Teubner-texte zur Mathematik, 8-18 (1987)
[2] Albrecht, P.: A new theoretical approach to RK methods. SIAM J. Numer. anal. 24, No. 2, 391-406 (1987) · Zbl 0617.65067
[3] Butcher, J. C.: The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods. (1987) · Zbl 0616.65072
[4] Coleman, J. P.: Numerical methods for y” = $f(x, y)$ via rational approximations for the cosine. IMA. J. Numer. anal. 9, 145-165 (1989) · Zbl 0675.65072
[5] Coleman, J. P.; Ixaru, L. Gr.: P-stability and exponential-Fitting methods for y” = $f(x, y)$. IMA J. Numer. anal. 16, 179-199 (1996) · Zbl 0847.65052
[6] Crisci, M. R.; Paternoster, B.; Russo, E.: Fully parallel Runge-Kutta Nyström methods for odes with oscillating solutions. Appl. numer. Math. 11, 143-158 (1993) · Zbl 0782.65090
[7] M.R. Crisci and B. Paternoster, Parallel Runge-Kutta Nyström methods, Ricerche Mat., to appear. · Zbl 0928.65087
[8] Dekker, K.; Verwer, J. G.: Stability of Runge-Kutta methods for stiff nonlinear differential equations. CWI monograph (1984) · Zbl 0571.65057
[9] Gautschi, W.: Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. math. 3, 381-397 (1961) · Zbl 0163.39002
[10] Hairer, E.; Norsett, S. P.; Wanner, G.: Solving ordinary differential equations I. Nonstiff problems. Springer series in computational mathematics 8 (1987) · Zbl 0638.65058
[11] Ixaru, L. Gr.: Numerical methods for differential equations and applications. (1984) · Zbl 0543.65047
[12] Ixaru, L. Gr.: Operations on oscillatory functions. Comput. phys. Comm. 105, 1-19 (1997) · Zbl 0930.65150
[13] Neta, B.; Ford, C. H.: Families of methods for ordinary differential equations based on trigonometric polynomials. J. comput. Appl. math. 10, 33-38 (1984) · Zbl 0529.65050
[14] Raptis, A. D.; Allison, A. C.: Exponential-Fitting methods for the numerical solution of the Schrödinger equation. Comput. phys. Comm. 44, 95-103 (1978)
[15] Raptis, A. D.; Simos, T. E.: A four-step phase-fitted method for the numerical integration of second order initial-value problems. Bit 31, 160-168 (1991) · Zbl 0726.65089
[16] Simos, T. E.; Dimas, E.; Sideridis, A. B.: A Runge-Kutta-Nyström method for the numerical integration of special second-order periodic initial-value problems. J. comput. Appl. math. 51, 317-326 (1994) · Zbl 0872.65066
[17] Sommeijer, B. P.: A note on a diagonally implicit Runge-Kutta-Nyström method. J. comput. Appl. math. 19, 395-399 (1987) · Zbl 0637.65065
[18] Stiefel, E.; Bettis, D. G.: Stabilization of cowell’s method. Numer. math. 13, 154-175 (1969) · Zbl 0219.65062
[19] Van Daele, M.; De Meyer, H.; Berghe, G. Vanden: A modified numerov integration method for general second order initial value problems. Internat. J. Comput. math. 40, 117-127 (1991) · Zbl 0738.65060
[20] Berghe, G. Vanden; De Meyer, H.; Vanthournout, J.: A modified numerov integration method for second order periodic initial-value problem. Internat. J. Comput. math. 32, 233-242 (1990) · Zbl 0752.65059
[21] Van Der Houwen, P. J.; Sommeijer, B. P.: Linear multistep methods with reduced truncation error for periodic initial-value problem. IMA J. Numer. anal. 4, 479-489 (1984) · Zbl 0566.65055
[22] Van Der Houwen, P. J.; Sommeijer, B. P.: Explicit Runge-$Kutta(-Nystr\"om)$ methods with reduced phase errors for computing oscillating solution. SIAM J. Numer. anal. 24, 595-617 (1987) · Zbl 0624.65058
[23] Van Der Houwen, P. J.; Sommeijer, B. P.: Phase-lag analysis of implicit Runge-Kutta methods. SIAM J. Numer. anal. 26, 214-228 (1989) · Zbl 0669.65055
[24] Van Der Houwen, P. J.; Sommeijer, B. P.: Diagonally implicit Runge-Kutta Nyström methods for oscillatory problems. SIAM J. Numer. anal. 26, 414-429 (1989) · Zbl 0676.65072
[25] Van Der Houwen, P. J.; Sommeijer, B. P.; Strehmel, K.; Weiner, R.: On the numerical integration of second order initial value problems with a periodic forcing function. Computing 37, 195-218 (1986) · Zbl 0589.65064