Functionally-fitted block methods for 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 ordinary differential equations. The basic theory for the proposed methods is established. First, we derive two sufficient conditions to ensure the existence of the functionally-fitted block methods, discuss their equivalence to collocation methods in a special case and independence on integration time for a set of separable basis functions. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the order of accuracy of the r-point functionally-fitted block method is at least r for general ordinary differential equations. Experimental results are conducted to demonstrate the validity of the functionally-fitted block methods.