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Phase-fitted and amplification-fitted two-step hybrid methods for \(y^{\prime\prime }=f(x,y)\). (English) Zbl 1141.65061

The author considers phase-fitted and amplitude-fitted two-step hybrid methods for the numerical integration of initial value problems for nonlinear ordinary differential equations featuring highly oscillatory solution behavior. The numerical approximation by standard methods often introduces a distorted frequency referred to as phase-lag.
The present paper provides a general theoretical framework for new phase-fitting techniques for two-step hybrid methods which are designed to overcome this problem as well as reduce the amplification error by means of introducing two new parameters. Convergence of the new methods is analyzed by a standard approach investigating consistency and stability.
As an example, a sixth-order method is constructed and the theoretical findings are validated by numerical comparisons with other well established methods.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
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[1] Anastassi, Z. A.; Simos, T. E., Special optimized Runge-Kutta methods for IVP’s with oscillating solutions, Internat. J. Modern Phys. C, 15, 1-15 (2004) · Zbl 1083.65066
[2] Anastassi, Z. A.; Simos, T. E., A dispersive-fitted and dissipative-fitted explicit Runge-Kutta method for the numerical solution of orbital problems, New Astronom., 10, 31-37 (2004) · Zbl 1060.65616
[3] Avdelas, G.; Simos, T. E., Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation, Phys. Rev. E, 62, 1375-1381 (2000)
[4] A.G. Bratsos, I. Th. Famelis, A. Kollias, Ch. Tsitouras, Phase-fitted Numerov type methods, Appl. Math. Comput., in press.; A.G. Bratsos, I. Th. Famelis, A. Kollias, Ch. Tsitouras, Phase-fitted Numerov type methods, Appl. Math. Comput., in press.
[5] Brusa, L.; Nigro, L., A one-step method for direct integration of structural dynamic equations, Internat. J. Numer. Methods Engrg., 15, 685-699 (1980) · Zbl 0426.65034
[6] Carpentieri, M.; Paternoster, B., Stability regions of one step mixed collocation methods for \(y'' = f(x, y)\), Appl. Numer. Math., 53, 201-212 (2005) · Zbl 1069.65092
[7] Chawla, M. M.; 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, 277-281 (1984) · Zbl 0565.65041
[8] Chawla, M. M.; Rao, P. S., An explicit sixth-order method with phase-lag of order eight for \(y'' = f(t, y)\), J. Comput. Appl. Math., 17, 365-368 (1987) · Zbl 0614.65084
[9] Chawla, M. M.; Sharma, S. R., Intervals of periodicity and absolute stability of explicit Nyström methods for \(y'' = f(x, y)\), BIT, 21, 455-464 (1981) · Zbl 0469.65048
[10] Coleman, J. P., Order conditions for a class of two-step methods for \(y'' = f(x, y)\), IMA J. Numer. Anal., 23, 197-220 (2003) · Zbl 1022.65080
[11] Coleman, J. P.; Duxbury, S. C., Mixed collocation methods for \(y'' = f(x, y)\), J. Comput. Appl. Math., 126, 47-75 (2000) · Zbl 0971.65073
[12] 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
[13] Franco, J. M., A class of explicit two-step hybrid methods for second-order IVPs, J. Comput. Appl. Math., 187, 41-57 (2006) · Zbl 1082.65071
[14] Franco, J. M.; Palacios, M., High-order P-stable multistep methods, J. Comput. Appl. Math., 30, 1-10 (1990) · Zbl 0726.65091
[15] Ixaru, L. Gr.; Rizea, M., Numerov method maximally adapted to the Schrödinger equation, J. Comput. Phys., 73, 306-324 (1987) · Zbl 0633.65131
[16] Ixaru, L. Gr.; Vanden Berghe, G., Exponential Fitting (2004), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0996.65075
[17] Lambert, J. D., Computational Methods in Ordinary Differential Equations (1973), Wiley: Wiley London · Zbl 0258.65069
[18] Lambert, J. D.; Watson, I. A., Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl., 18, 189-202 (1976) · Zbl 0359.65060
[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, 25-46 (1999) · Zbl 1306.65229
[20] Papageorgiou, G.; Tsitouras, Ch.; Famelis, I., Explicit Numerov type methods for second order IVPs with oscillating solutions, Internat. J. Modern Phys. C, 12, 657-666 (2001)
[21] Paternoster, B., A phase-fitted collocation based Runge-Kutta-Nyström method, Appl. Numer. Math., 35, 339-355 (2000) · Zbl 0979.65063
[22] 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
[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, 135-140 (1991) · Zbl 0747.65061
[24] Simos, T. E., Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation, Comput. & Chem., 23, 439-446 (1999)
[25] Tsitouras, Ch., Explicit two-step methods for second-order linear IVPs, Comput. Math. Appl., 43, 943-949 (2002) · Zbl 1050.65071
[26] Tsitouras, Ch., Explicit Numerov type methods with reduced number of stages, Comput. Math. Appl., 45, 37-42 (2003) · Zbl 1035.65078
[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, 115-130 (2005) · Zbl 1196.65117
[28] Van de Vyver, H., Frequency evaluation for exponentially-fitted Runge-Kutta methods, J. Comput. Appl. Math., 184, 442-463 (2005) · Zbl 1077.65082
[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, 278-285 (2006) · Zbl 1187.65078
[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, 309-318 (2006) · Zbl 1086.65073
[31] Van de Vyver, H., A fourth-order symplectic exponentially fitted integrator, Comput. Phys. Comm., 174, 255-262 (2006) · Zbl 1196.37122
[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, 663-675 (2006) · Zbl 1107.82304
[33] van der Houwen, P. J.; Sommeijer, B. P., Explicit Runge-Kutta(-Nyström) methods with reduced phase errors for computing oscillating solutions, SIAM J. Numer. Anal., 24, 595-617 (1987) · Zbl 0624.65058
[34] Van Dooren, R., Stabilization of Cowell’s classical finite difference methods for numerical integration, J. Comput. Phys., 16, 186-192 (1974) · Zbl 0294.65042
[35] Vigo-Aguiar, J.; Simos, T. E.; 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, 561-567 (2004) · Zbl 1041.65058
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