TITLE

A three-stage explicit two-step Runge-Kutta-Nystro¨m method for solving second-order ordinary differential equations

AUTHOR(S)
Md Ariffin, Latifah; Senu, Norazak; Suleiman, Mohamed
PUB. DATE
April 2013
SOURCE
AIP Conference Proceedings;Apr2013, Vol. 1522 Issue 1, p323
SOURCE TYPE
Conference Proceeding
DOC. TYPE
Article
ABSTRACT
A three-stage explicit two-step Runge-Kutta-Nystro¨m (TSRKN) method is developed for the numerical integration of special second-order ordinary differential equations. Algebraic order conditions of the method are obtained and fourth-order method is derived. The second-order initial value problems of ordinary differential equations (ODEs) are solved directly using TSRKN and Runge-Kutta-Nystro¨m (RKN) methods. The problems were then reduced to first-order system when solved by Runge-Kutta (RK) method. Numerical comparison of this new method with the existing RK and RKN methods of the same order using constant step size are carried out to illustrate its efficiency and it shows that the new method has clear advantage in terms of function evaluation.
ACCESSION #
87085988

 

Related Articles

  • Numerical solution of first order initial value problem using 7-stage tenth order Gauss-Kronrod-Lobatto IIIA method. Ying, Teh Yuan; Yaacob, Nazeeruddin // AIP Conference Proceedings;Apr2013, Vol. 1522 Issue 1, p183 

    In this paper, a new implicit Runge-Kutta method which based on a 7-point Gauss-Kronrod-Lobatto quadrature formula is developed. The resulting implicit method is a 7-stage tenth order Gauss-Kronrod-Lobatto IIIA method, or in brief as GKLM(7,10)-IIIA. GKLM(7,10)-IIIA requires seven function of...

  • Solving the order reduction phenomenon in variable step size quasi-consistent Nordsieck methods. Kulikov, G. // Computational Mathematics & Mathematical Physics;Nov2012, Vol. 52 Issue 11, p1547 

    The phenomenon is studied of reducing the order of convergence by one in some classes of variable step size Nordsieck formulas as applied to the solution of the initial value problem for a first-order ordinary differential equation. This phenomenon is caused by the fact that the convergence of...

  • A three-stage dispersion and dissipation of order infinity Runge-Kutta-Nyström method for periodic IVPs. Wing, Moo Kwong; Senu, Norazak; Suleiman, Mohamed; Ismail, Fudziah // AIP Conference Proceedings;Sep2013, Vol. 1557 Issue 1, p263 

    In this paper, a new three-stage Runge-Kutta-Nyström (RKN) method with two variable coefficients is constructed. The new method is based on Garcia's RKN method of algebraic order four. By using the idea of dispersion and dissipation of order infinity, a new method for solving second-order...

  • FOURTH ORDER 4-STAGES IMPROVED RUNGE-KUTTA METHOD WITH MINIMIZED ERROR NORM. Rabiei, Faranak; Ismail, Fudziah // AIP Conference Proceedings;2014, Vol. 1613, p153 

    In this paper the improved Runge-Kutta method of order four with 4-stages for solving first order ordinary differential equation is proposed. The method is based on classical Runge-Kutta (RK) method also can be considered as special class of two-step method. Here, the coefficients of the method...

  • Composite Group of Explicit Runge-Kutta Methods. Fatin Nadiah Abd Hamid; Faranak Rabiei; Fudziah Ismail // AIP Conference Proceedings;2016, Vol. 1739 Issue 1, p020057-1 

    In this paper, the composite groups of Runge-Kutta (RK) method are proposed. The composite group of RK method of third and second order, RK3(2) and fourth and third order RK4(3) base on classical Runge-Kutta method are derived. The proposed methods are two-step in nature and have less number of...

  • Contractivity-preserving explicit Hermite—Obrechkoff ODE solver of order 13. Nguyen-Ba, Truong; Desjardins, Steven J.; Sharp, Philip W.; Vaillancourt, Rémi // Celestial Mechanics & Dynamical Astronomy;Dec2013, Vol. 117 Issue 4, p423 

    A new optimal, explicit, Hermite–Obrechkoff method of order 13, denoted by HO(13), that is contractivity-preserving (CP) and has nonnegative coefficients is constructed for solving nonstiff first-order initial value problems. Based on the CP conditions, the new 9-derivative HO(13) has...

  • Approximation methods for second order nonlinear polylocal problems. Pop, Daniel N.; Trîmbiţaş, Radu T. // Studia Universitatis Babes-Bolyai, Mathematica;2011, Vol. 56 Issue 2, p515 

    Consider the problem: y′ (x) + f(x, y) = 0, x ε [0, 1] y(a) = α y(b) = β, a, b ε (0, 1). This is not a two-point boundary value problem since a, b 2 (0, 1). It is possible to solve this problem by dividing it into the three problems: a two-point boundary value problem (BVP) on...

  • A special class of continuous general linear methods. Yakubu, D. G.; Kwami, A. M.; Ahmed, M. L. // Computational & Applied Mathematics;2012, Vol. 31 Issue 2, p259 

    We consider the construction of a class of numerical methods based on the general matrix inverse [14] which provides continuous interpolant for dense approximations (output). Their stability properties are similar to those for Runge-Kutta methods. These methods provide a unifying scope for many...

  • Global error estimation with adaptive explicit Runge-Kutta methods. CALVO, M.; HIGHAM, D. J.; MONTIJANO, J. I.; RÁNDAZ, L. // IMA Journal of Numerical Analysis;1996, Vol. 16 Issue 1, p47 

    Users of locally-adaptive software for initial value ordinary differential equations are likely to be concerned with global errors. At the cost of extra computation, global error estimation is possible. Zadunaisky's method and ‘solving for the error estimate’ are two techniques that...

Share

Read the Article

Courtesy of THE LIBRARY OF VIRGINIA

Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics