A new diagonally implicit Runge-Kutta-Nyström method for periodic IVPs
Pages 679 - 688
Abstract
A new diagonally implicit Runge-Kutta-Nyström (RKN) method is developed for the integration of initial-value problems for second-order ordinary differential equations possessing oscillatory solutions. Presented is a method which is three-stage fourth-order with dispersive order six and 'small' principal local truncation error terms and dissipation constant. The analysis of phase-lag, dissipation and stability of the method are also given. This new method is more efficient when compared with current methods of similar type for the numerical integration of second-order differential equations with periodic solutions, using constant step size.
References
[1]
D. I. Okunbor and R. D. Skeel, Canonical Runge-Kutta-Nyström methods of order five and six, J. Comput. Appl. Math., vol. 51, 1994, pp. 375-382.
[2]
E. Hairer and G. Wanner, A Theory for Nystrom Methods, Numer. Math., vol. 25, 1975, pp. 383-400.
[3]
E. Stiefel and D.G. Bettis, Stabilization of Cowell's methods, Numer. Math, vol. 13, 1969, pp. 154-175.
[4]
G. Papageorgiou, I. Th. Famelis and Ch. Tsitouras, A P-stable single diagonally implicit Runge-Kutta-Nyström method, Numerical Algorithms, vol. 17, 1998, pp. 345- 353.
[5]
I. Gladwell and R. Thomas, Damping and phase analysis of some methods for solving second order ordinary differential equations, Internat. J. Numer. Methods Engrg., vol. 19, 1983, pp. 495-503.
[6]
J.C. Butcher, Numerical Methods for Ordinary Differential Equations, Wiley & Sons Ltd., England, 2003.
[7]
J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial-value problems, J. Inst. Maths Applics, vol. 18, 1976, pp. 189-202.
[8]
J. D. Lambert, Numerical Methods for ordinary Differential Systems-The Initial Value Problem, Wiley & Sons Ltd., England, 1991.
[9]
J. R. Dormand, Numerical Methods for Differentials Equations, CRC Press, Inc, Florida, 1996.
[10]
L. Bursa and L. Nigro, A one-step method for direct integration of structural dynamic equations, Internat. J. Numer. Methods Engrg, vol. 15, 1980, pp. 685-699.
[11]
M. M. Chawla and S. R Sharma, Intervals of periodicity and absolute stability of explicit Nystrom methods for y″ = f (x,y), BIT, vol. 21, 1981, pp. 455-469.
[12]
P. J. van der Houwen and B. P. Sommeijer, Explicit Runge-Kutta(-Nyström) methods with reduced phase errors for computing oscillating solutions, SIAM J. Numer. Anal., vol. 24, no. 3, 1987, pp. 595-617.
[13]
P. J. van der Houwen and B. P. Sommeijer, "Diagonally implicit Runge-Kutta(-Nyström) methods for oscillatory problems," SIAM J. Numer. Anal., vol. 26, no. 2, 1989, pp. 414- 429.
[14]
P. W. Sharp, J. M. Fine and K. Burrage, Two-stage and Three-stage Diagonally Implicit Runge-Kutta-Nyström Methods of Orders Three and Four, J. Of Numerical Analysis, vol. 10, 1990, pp. 489-504.
[15]
T. E. Simos, E. Dimas, A.B. Sideridis, A Runge-Kutta-Nyström method for the integration of special second-order periodic initial-value problems, J. Comput. Appl. Mat., vol 51, 1994, pp. 317-326.
[16]
W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polinomials, Numer. Math. vol. 3, 1961, pp. 381-397.
[17]
S. O. Imoni, F. O. Otunta and T. R. Ramamohan, Embedded implicit Runge-Kutta-Nyström method for solving second-order differential equations, International Journal of Computer Mathematics, vol. 83, no. 11, 2006, pp. 777-784.
[18]
N. Senu, M. Suleiman and F. Ismail. (2009, June). An embedded explicit Runge-Kutta-Nyström method for solving oscillatory problems. 80(1). Available: stacks.iop.org/PhysScr/80/015005
[19]
R. C. Allen, Jr. and G. M. Wing, An invariant imbedding algorithm for the solution of inhomogeneous linear two-point boundary value problems, J. Computer Physics, vol. 14, 1974, pp. 40-58.
[20]
B.P. Sommeijer, A note on a diagonally implicit Runge-Kutta-Nystrom method, J. Comp. Appl. Math., vol 19, 1987, pp. 395- 399.
[21]
Hairer, S., Norsett, P. & Wanner, G. Solving Ordinary Differential Equations I, Nonstiff Problems, 2nd ed. Berlin, Heidelberg, New-York: Springer-Verlag. 1993.
[22]
J. R. Dormand, M. E. A. El-Mikkawy and P. J. Prince, Families of Runge-Kutta-Nyström Formulae, IMA J. Numer. Anal., vol. 7, 1987, pp. 235-250.
[23]
Al-Khasawneh, R.A., Ismail, F., Suleiman, M., Embedded diagonally implicit Runge-Kutta-Nyström 4(3) pair for solving special second-order IVPs. Appl. Math. Comp., vol. 190, 2007, 1803-1814.
[24]
F. Ismail, N. I. Che Jawias, M. Suleiman and A. Jaafar, Singly Diagonally Implicit fifth order five-stage Runge-Kutta method for Linear Ordinary Differential Equations, Proceedings of the 8th WSEAS International Conference on Applied Computer and Applied Computational Science, ISSN: 1790-5117, 82- 86.
[25]
N. Razali, R. Ahmad, M. Darus, A. Rambely, Fifth-Order Mean Runge-Kutta Methods Applied to the Lorenz System, Proceedings of the 13th WSEAS International Conference on APPLIED MATHEMATICS (MATH'08), ISSN: 1790-2769, 333-338.
[26]
F. Ismail, N. I. Che Jawias, M. Suleiman and A. Jaafar, Solving Linear Ordinary Differential Equations using Singly Diagonally Implicit Runge-Kutta fifth order five-stage method, WSEAS Transactions On Mathematics, vol 8, No. 8, 2009, pp. 393-402.
[27]
J.R. Dormand, M.E.A. El-Mikkawy and P.J. Prince, Families of Runge-Kutta-Nyystrom Formulae, IMA J. Numer. Anal., vol. 7, 1987, pp. 235-250.
[28]
M. Podisuk and S. Phummark, Newton-Cotes Formula in Runge-Kutta Method, WSEAS Transactions On Mathematics, vol 4, no. 1, 2005, pp. 18-23.
- A new diagonally implicit Runge-Kutta-Nyström method for periodic IVPs
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Published: 01 September 2010
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