Abstract
This paper proposes and investigates a special class of explicit Runge-Kutta-Nyström (RKN) methods for problems in the form y′′(x)=f(x,y,y′) including third derivatives and denoted as STDRKN. The methods involve one evaluation of second derivative and many evaluations of third derivative per step. In this study, methods with two and three stages of orders four and five, respectively, are presented. The stability property of the methods is discussed. Numerical experiments have clearly shown the accuracy and the efficiency of the new methods.
Highlights
In this article, we are interested in initial value problems (IVPs) of second-order ordinary differential equations (ODEs): y (x) = f (x, y (x), y (x)), y (x0) = α, (1)y (x0) = β, x ∈ [x0, xend], where y ∈ RN, f : R × RN × RN → RN are continuous vector valued functions
This paper proposes and investigates a special class of explicit Runge-Kutta-Nystrom (RKN) methods for problems in the form y(x) = f(x, y, y) including third derivatives and denoted as STDRKN
We are interested in initial value problems (IVPs) of second-order ordinary differential equations (ODEs): y (x) = f (x, y (x), y (x)), y (x0) = α, (1)
Summary
We are interested in initial value problems (IVPs) of second-order ordinary differential equations (ODEs): y (x) = f (x, y (x) , y (x)) , y (x0) = α, (1). Y (x0) = β, x ∈ [x0, xend] , where y ∈ RN, f : R × RN × RN → RN are continuous vector valued functions. This type of problems arises naturally in many applied science fields such as the Kepler problems in celestial mechanics, quantum physics, and Newton’s second law in classical mechanics (see Dormand [1], Hairer et al [2], and Kristensson [3]).
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