Linear Shooting with High-Order Diagonally Implicit Runge-Kutta-Nyström Methods for Types of Second-Order Linear Boundary Value Problems with Applications

<ns3:p>Background Second-order linear boundary value problems arise in many applications of science and engineering and are commonly treated by reducing them to first-order systems, which increases the computational cost. Direct high-order methods that preserve the second-order structure are therefore of interest. Methods This study proposes fixed-step, high-order diagonally implicit Runge–Kutta–Nyström (DIRKN) methods for the direct numerical solution of special second-order linear boundary value problems. Two schemes, DIRKN(3,4) and DIRKN(4,5), are constructed and combined with a linear shooting technique to handle both Dirichlet and Neumann boundary conditions without transforming the problem into a first-order system. Results The performance of the proposed methods is evaluated using benchmark test problems and application models, including rod heat conduction and reaction–diffusion equations. Numerical results demonstrate that both schemes achieve high accuracy and good efficiency, with DIRKN(4,5) generally producing smaller errors for a comparable number of function evaluations. A linear stability analysis shows absolute stability intervals of (−1.96, 0) for DIRKN(3,4) and (−7.24, 0) for DIRKN(4,5). Conclusions The proposed DIRKN methods provide accurate and efficient solvers for second-order linear boundary value problems with different types of boundary conditions, while preserving the original problem structure and reducing computational cost.</ns3:p>