Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1111.65066
Imoni, S.O.; Otunta, F.O.; Ramamohan, T.R.
Embedded implicit Runge-Kutta Nyström method for solving second-order differential equations.
(English)
[J] Int. J. Comput. Math. 83, No. 11, 777-784 (2006). ISSN 0020-7160; ISSN 1029-0265/e

Summary: An embedded diagonally implicit Runge-Kutta Nyström (RKN) method is constructed for the integration of initial-value problems for second-order ordinary differential equations possessing oscillatory solutions. This embedded method is derived using a three-stage diagonally implicit RKN method of order four within which a third-order three stage diagonally implicit RKN method is embedded. We demonstrate how this system can be solved, and by an appropriate choice of free parameters, we obtain an optimized $RKN(4,3)$ embedded algorithm. We also examine the intervals of stability and show that the method is strongly stable within an appropriate region of stability and is thus suitable for oscillatory problems by applying the method to the test equation $y^{\prime \prime}=-\omega^{2}y,\omega >0$. Necessary and sufficient conditions are given for this method to possess non-vanishing intervals of periodicity, for the fourth-order method. Finally, we present the coefficients of the method optimized for small truncation errors. This new scheme is likely to be efficient for the numerical integration of second-order differential equations with periodic solutions, using adaptive step size.
MSC 2000:
*65L06 Multistep, Runge-Kutta, and extrapolation methods
65L05 Initial value problems for ODE (numerical methods)
34A34 Nonlinear ODE and systems, general
65L20 Stability of numerical methods for ODE
65L50 Mesh generation and refinement (ODE)

Keywords: initial value problems; diagonally implicit; second-order equations; step-size control; numerical examples; oscillatory solutions; stability

Highlights
Master Server