Insight into Reformulation of an A-stable Runge-Kutta (RK) Method as an Implicit Runge-Kutta-Nystrom (RKN) Method for Directly Solving Second-order ODEs

This work intends to contribute to the growing literature on the direct numerical solution of special and general second-order initial value problems (IVPs) of ordinary differential equations (ODEs) by reformulating an A-stable Runge-Kutta (RK) method as a type of implicit Runge–Kutta–Nystrӧm methods (RKN). Unlike traditional methods that reduce second-order IVPs of ODEs to equivalent systems of first-order IVPs, the approach in this work preserves the original problem structure while also benefiting from the sixth-order accuracy and A-stability of the GLRK method. Using an extended Butcher array, the resulting RKN coefficients are obtained explicitly, ensuring they meet the consistency and order conditions. The linear stability analysis reveals a broad stability region, which is necessary for handling stiff and oscillatory systems. Application of the method to numerical examples shows that it offers better accuracy than most existing methods, while maintaining a similar computational cost. This reformulation strategy paves the way for deriving high-order A-stable RKN methods based on existing RK schemes.