A unified approach for the development of k-step block Falkner-type methods for solving general second-order initial-value problems in ODEs

A family of k-step block multistep methods where the main formulas are of Falkner type is proposed for the direct integration of the general second order initial-value problem where the differential equation is of the general form y″=f(x,y,y′). The two main Falkner formulas and the additional ones to complete the block procedure are obtained from a continuous approximation derived via interpolation and collocation at k+1 points. The main characteristics of the methods are discussed through their formulation in vector form. Each method is formulated as a group of 2k simultaneous formulas over k non-overlapping intervals. In this way, the method produces the approximation of the solution simultaneously at k points on these intervals. As in other block methods, there is no need of other procedures to provide starting approximations, and thus the methods are self-starting (sharing this advantage of Runge-Kutta methods). The resulting family is efficient and competitive compared with other existing methods in the literature, as may be seen from the numerical results.