Explicit Runge–Kutta–Nyström-Type Schemes for Fourth-Order Systems y(4)=f (x, y, y′)

This work addresses the numerical solution of fourth-order initial value problems of the form y(4)=f(x,y,y′), extending the capabilities of standard Runge–Kutta–Nyström (RKN) methods which are typically limited to y(4)=f(x,y). Problems of this type arise naturally in structural and vibroacoustic dynamics, where velocity-dependent damping and coupling effects are essential for realistic modeling. Despite their practical importance, efficient explicit schemes that preserve the fourth-order structure while allowing derivative dependence remain limited. We generally present an explicit s-stage method that incorporates the first derivative into the internal stage approximations, necessitating the introduction of a new matrix parameter D in the order conditions. We successfully derive the algebraic order conditions for this extended method up to the seventh algebraic order. A particular pair of orders 6(4) is constructed at an effective cost of only four stages per step in contrast to eight function evaluations required in conventional RK pairs. This reduction in effective stage cost, together with the direct treatment of derivative-dependent terms, constitutes a structural and computational distinction from existing Runge–Kutta and RKN approaches. To demonstrate the physical relevance of the proposed solvers, we examine coupled fourth-order models arising in structural and vibroacoustic dynamics, including viscoelastic beam systems with aerodynamic (velocity-proportional) damping and structure–acoustic interaction in a thin-walled duct. These examples illustrate the capability of the method to handle coupled dynamics with derivative-dependent damping and source terms that are central to realistic modeling of such systems. On these representative problems, the proposed pair clearly and decisively outperforms existing Runge–Kutta pairs from the current literature, achieving substantially higher accuracy for the same computational effort. The results indicate that explicit fourth-order Nyström-type schemes with derivative-aware internal stages provide both a theoretical extension of classical RKN theory and measurable efficiency gains, offering a competitive alternative to reduction-based first-order formulations for velocity-dependent fourth-order systems.