On the full design of third-order BN-stable implicit integrators for nonlinear dynamics

The numerical simulation of nonlinear structural dynamics remains a challenging task in engineering practice. Conventional A-stable time integration schemes, though widely used, may diverge in highly nonlinear problems and yield unusable results. To overcome this limitation, BN-stable three-substep composite time integration schemes are investigated to ensure robust stability in nonlinear regimes. A comprehensive analysis is conducted for three-stage, third-order singly diagonally implicit Runge–Kutta schemes, with the three-substep composite formulations cast into the SDIRK3 class via identical displacement and velocity iterative forms, leading to the identification of all algebraically stable members and their extension into a family with flexible, controllable numerical dissipation tailored to structural dynamics. It is proved that any third-order BN-stable SDIRK3 scheme is necessarily dissipative, and a non-dissipative member does not exist within this class. Two optimal BN-stable schemes are selected from the resulting family to facilitate computations for large-scale structural systems. Strategies for acceleration output are also developed, addressing a classical shortcoming of Runge–Kutta formulations. The proposed family further includes a fourth-order member and a parameter setting that minimizes period errors. Nonlinear benchmarks confirm unconditional stability and tunable dissipation, and both a one-dimensional wave-propagation problem and simulations of a complex folding rudder with joint clearance demonstrate the applicability of the new methods to real structures.