zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
65L20Stability and convergence of numerical methods for ODE
65L06Multistep, Runge-Kutta, and extrapolation methods
Full Text: DOI
[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)
[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.
[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.; Berghe, G. Vanden: Exponential Fitting. (2004) · Zbl 1105.65082
[17] Lambert, J. D.: Computational methods in ordinary differential equations. (1973) · 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 -- $Kutta( -- Nystr\"om)$ 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