### An ecient family of second derivative Runge-Kutta collocation methods for oscillatory systems

#### Abstract

An ecient family of high-order second derivative Runge-Kutta collocation methods is de-

rived for the numerical solution of oscillatory systems. The approach uses polynomial inter-

polation and collocation techniques to construct continuous schemes which were evaluated

at both step and o-step points to obtain hybrid formulae. The hybrid formulae can be

applied simultaneously as block methods for moving the integration process forward at a

time, if desired. The block methods based on hybrid formulation can also be converted to

second derivative Runge-Kutta collocation methods. The stability properties and order of

accuracy of the methods are studied. They can also be implemented easily since they are

collocation methods and provide a high order of accuracy. The methods were illustrated by

the applications to some test problems of oscillatory system found in the literature and the

numerical results obtained conrm the accuracy and eciency of the methods.

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