Assessments of dissipative structure-dependent integration methods

Two $Chang-{\alpha}$ dissipative family methods and two $KR-{\alpha}$ family methods were developed for time integration recently. Although the four family methods are in the category of the dissipative structure-dependent integration methods, their performances may be drastically different due to the detrimental property of weak instability or overshoot for the two $KR-{\alpha}$ family methods. This weak instability or overshoot will result in an adverse overshooting behavior or even numerical instability. In general, the four family methods can possess very similar numerical properties, such as unconditional stability, second-order accuracy, explicit formulation and controllable numerical damping. However, the two $KR-{\alpha}$ family methods are found to possess a weak instability property or overshoot in the high frequency responses to any nonzero initial conditions and thus this property will hinder them from practical applications. Whereas, the two $Chang-{\alpha}$ dissipative family methods have no such an adverse property. As a result, the performances of the two $Chang-{\alpha}$ dissipative family methods are much better than for the two $KR-{\alpha}$ family methods. Analytical assessments of all the four family methods are conducted in this work and numerical examples are used to confirm the analytical predictions.