Traditionally, numerical time-integration algorithms for linear dynamics are analyzed only with reference to the homogenous part of the solution [1]. Although greatly simpli"ed and able to provide useful indications about the error evolution, this accuracy analysis is not complete. The accuracy analysis of the forcing term was performed in the past only in a few studies [2], usually limited to the investigation of the local truncation error.
A more complete approach for the accuracy analysis was presented by Preumont [3]. The time-integration operators are treated as digital recursive "lters, so that the transfer functions of the discretized equations can be derived. The comparison between the numerical and the exact transfer functions provides information regarding the algorithm behavior with a forcing term and allows one to detect possible spurious resonance conditions.
This approach was adopted for the analysis of non-dissipative second and higher order algorithms in reference [4]. Higher order methods showed a better performance than traditional second order ones in the quasi-resonance condition. This analysis was performed only for conservative methods. It is well known that when a structural system is discretized by the "nite-element methodology, only the vibration modes associated with the lower frequencies are meaningful [1]. Therefore, it is desirable that the time-stepping method possess high-frequency dissipation in order to damp out the inaccurate response of the high-frequency modes. The HHT-method [1], characterized by second order accuracy, is one of the most popular dissipative algorithms for structural dynamics. However, due to its relatively low order accuracy, a non-negligible error may result in long-term simulations.
In recent years, extensive research has been conducted on dissipative higher order methods (see for example the surveys given by Fung [5,6]). The main feature of these methods is the coupling of high-frequency dissipation and low rate of error growth, which is particularly advantageous in long-term simulations. In particular, the time discontinuous Galerkin (TDG) methodology of time discretization leads to very attractive higher order dissipative methods. Although at an increased computational cost, TDG algorithms show an improved performance over traditional second order methods [7].
In this work, the widely used HHT-scheme and a TDG method are compared, with reference to the behavior near the resonance condition by means of the approach introduced by Preumont [3]. It will be shown that the TDG algorithm has an improved performance in such a condition compared to the HHT-method. Therefore, the TDG algorithm should be used in analyses near the resonance condition when an algorithm with high-frequency modes dissipation is required.