Two-step RKN Direct Method for Special Second-order Initial and Boundary Value Problems

In this study, a class of direct numerical integrators for solving special second-order ordinary differential equations (ODEs) is proposed and studied. The method is multistage and multistep in nature. This class of integrators is called "two-step Runge-Kutta-Nystrom", denoted by TSRKN. The direct approach to higher-order ODEs is desirable to avoid tedious computational work caused by converting the higherorder ODEs into the system of first-order equations. The order conditions for the TSRKN are derived using Taylors series expansion and according to the order conditions, a three-stage TSRKN method which is convergent of order four is constructed. The convergence analysis of the method is discussed and the performance of the newly derived method is compared with existing methods. The numerical results show the superiority of the TSRKN method in terms of number of function evaluations and demonstrate that the TSRKN can also be used to solve linear second-order boundary value problems (BVPs) since Runge-Kutta-Nystrom (RKN) approach is practically used to only solve higher-order initial value problems (IVPs) directly.