Functionally-fitted block methods for second order ordinary differential equations

Functionally-fitted methods generalize collocation techniques to integrate exactly a chosen set of linearly independent functions. In this paper, we propose a new type of functionally-fitted block methods for initial value problems of second order ordinary differential equations. The basic theory for the proposed methods is established. First, we present a sufficient condition for the existence of the functionally-fitted block methods. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the r-point functionally-fitted block method is convergent of order at least r for second order ordinary differential equations. Numerical experiments are conducted to demonstrate the validity and efficiency of the functionally-fitted block methods by comparison with some existing numerical methods.