Embedded Singly Diagonally Implicit Runge-Kutta –Nystrom Method Order 5(4) for the Integration of Special Second Order ODEs
In this paper a new embedded Singly Diagonally
Implicit Runge-Kutta Nystrom fourth order in fifth order method for
solving special second order initial value problems is derived. A
standard set of test problems are tested upon and comparisons on the
numerical results are made when the same set of test problems are
reduced to first order systems and solved using the existing
embedded diagonally implicit Runge-Kutta method. The results
suggests the superiority of the new method.
[1] Sharp and Fine, Some Nystrom pairs for the general second order
initial-value Problem, Journal of Computational and Applied
Mathematics 42: 279-291.(1992).
[2] J. R. Dormand , M. E. A. El-Mikkawy and P. J. Prince, Families of
Runge-Kutta Nystrom Formula, IMA Journal of Numerical Analysis, 7:
235-250. (1987).
[3] M. E. A. El-Mikkawy and R. El- Desouky, A new optimized non-FSAL
embedded Runge-Kutta-Nystrom algorithm of orders 6 and 4 in six
stages, Applied Mathematics and Computation 145 : 33-43, (2003).
[4] G. Papageorgiou., I. Th. Famwlis and Ch. Tsitouras, A P-stable singly
diagonally implicit Runge-Kutta-Nystrom method. Numerical Algorithm
17: 345-353.(1998).
[5] Hairer, E. and Wanner, G. (1987), Solving Ordinary Differential
Equations I, Springer-Verlag Berlin.
[6] J. R. Dormand. Numerical Methods for Differential Equations A
Computational Approach, CRC Press London. 1996, p 57.
[7] M.M. Chawla and P. S. Rao, High-accuracy P-stable methods for
y'' = f (t, y), IMA J. Numer.Anal.5:215-220 (1985).
[8] 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 14: 40-58.(1974).
[9] J. C. Butcher and D. J. Chen,. A new Type of Singly-implicit Runge-
Kutta method, Applied Numerical Mathematics,, 34: 179-188. (2000).
[1] Sharp and Fine, Some Nystrom pairs for the general second order
initial-value Problem, Journal of Computational and Applied
Mathematics 42: 279-291.(1992).
[2] J. R. Dormand , M. E. A. El-Mikkawy and P. J. Prince, Families of
Runge-Kutta Nystrom Formula, IMA Journal of Numerical Analysis, 7:
235-250. (1987).
[3] M. E. A. El-Mikkawy and R. El- Desouky, A new optimized non-FSAL
embedded Runge-Kutta-Nystrom algorithm of orders 6 and 4 in six
stages, Applied Mathematics and Computation 145 : 33-43, (2003).
[4] G. Papageorgiou., I. Th. Famwlis and Ch. Tsitouras, A P-stable singly
diagonally implicit Runge-Kutta-Nystrom method. Numerical Algorithm
17: 345-353.(1998).
[5] Hairer, E. and Wanner, G. (1987), Solving Ordinary Differential
Equations I, Springer-Verlag Berlin.
[6] J. R. Dormand. Numerical Methods for Differential Equations A
Computational Approach, CRC Press London. 1996, p 57.
[7] M.M. Chawla and P. S. Rao, High-accuracy P-stable methods for
y'' = f (t, y), IMA J. Numer.Anal.5:215-220 (1985).
[8] 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 14: 40-58.(1974).
[9] J. C. Butcher and D. J. Chen,. A new Type of Singly-implicit Runge-
Kutta method, Applied Numerical Mathematics,, 34: 179-188. (2000).
@article{"International Journal of Engineering, Mathematical and Physical Sciences:58880", author = "Fudziah Ismail", title = "Embedded Singly Diagonally Implicit Runge-Kutta –Nystrom Method Order 5(4) for the Integration of Special Second Order ODEs", abstract = "In this paper a new embedded Singly Diagonally
Implicit Runge-Kutta Nystrom fourth order in fifth order method for
solving special second order initial value problems is derived. A
standard set of test problems are tested upon and comparisons on the
numerical results are made when the same set of test problems are
reduced to first order systems and solved using the existing
embedded diagonally implicit Runge-Kutta method. The results
suggests the superiority of the new method.", keywords = "Runge-Kutta Nystrom, Special second orderproblems.", volume = "2", number = "2", pages = "124-5", }