A four-stage implicit Runge-Kutta-Nyström method with variable coefficients for solving periodic initial value problems

Abstract

A new implicit Runge-Kutta-Nyström method with variable coefficients is developed for solving the periodic initial value problem of the differential equation y" = f (t, y). The proposed method, whose coefficients are functions of the frequency and the stepsize, integrates exactly the equation, if the solution is a periodic function with a single Fourier component and the frequency is known. On the other hand, the order of accuracy of the method is shown to be 4 for the case that an estimated frequency, instead of the exact one, is applied to evaluate the coefficients, as well as for that the solution is non-periodic.

A functional fitting Runge-Kutta-Nyström method with variable coefficients

“…for y(t) = cos(wt), sin(wt), so that these methods are trigonometric RKN methods in the terminology of Ozawa [7]. For the coefficients of these methods, we prepare not only the analytic expressions but also the power series expansions, which are particularly useful when computing the coefficients numerically for small wh to avoid the cancellation.…”

“…The same idea that of Gautschi has been applied to Runge-Kutta(-Nyström) methods and several authors have proposed trigonometric-or exponential-fitting methods (e.g. [7,9,14]). Another type of variable coefficients methods based on mixed interpolation or collocation, which is the interpolation or collocation by the functions of the form Ek=1 ak t k-1 + Et -1(ßa cos(rit) +'y sin(rlt)), have been proposed by several authors (e.g.…”

Different approaches in GLONASS orbit computation from broadcast ephemeris

“…Runge-Kutta (RK) methods are one of the well-known numerical methods for solving differential equations (Kosti et al, 2009;Ozawa, 1999;Sermutlu, 2004), while 4 th order Runge-Kutta method is recommended to solve equations of satellite motion by GLONASS Interface Control Document (ICD-GLONASS, 2008).…”

A functionally fitted three-stage explicit singly diagonally implicit Runge-Kutta method

“…[2], [7], [10], [11], [13], [14]). To be able to fit Runge-Kutta (-Nystrom) methods to any desired functions, Ozawa [8], [9] has recently developed a technique to construct the RungeKutta (-Nystrom) method that is exact on the linear space of given functions.…”