Stability and accuracy analysis of classical time-stepping algorithms in a unified characteristic-polynomial framework

Absztrakt

This paper presents a unified spectral framework for evaluating the stability and accuracy of classical time-stepping algorithms via the characteristic polynomials of their approximation operators. The Central Difference Method and the Wilson–θ scheme are analysed using the undamped single-degree-of-freedom oscillator, enabling numerical effects to be distinguished from physical behaviour. Stability is established through the Jury criterion, while numerical damping and phase error are quantified from the dominant eigenvalues. The Central Difference Method is conditionally stable and amplitude-preserving but exhibits a second-order negative phase error, whereas the Wilson–θ Method achieves unconditional stability at the cost of fourth-order numerical dissipation and a positive second-order phase error. The proposed framework provides a transparent basis for comparing integration schemes and highlights the trade-off between unconditional stability and long-time accuracy.