An efficient family of high-order second derivative

Runge-Kutta collocation methods is derived for the numerical

solution of oscillatory systems.

The approach uses polynomial interpolation and collocation techniques to construct continuous schemes which were evaluated at both step and off-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 confirm the accuracy and efficiency of the methods.

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