The Hilber–Hughes–Taylor-α (HHT-α) method compared with an implicit Runge–Kutta for second-order systems

We will consider the Hilber–Hughes–Taylor-α (HHT-α ) method to solve periodic second-order initial value problems arising in, e.g. mechanics. We will consider the analysis of the method when applied to such problems. Second-order convergence is theoretically demonstrated and numerically illustrated. In addition, we will consider the efficient implementation of an implicit fourth-order Runge–Kutta scheme. Numerical details and examples will also be presented to demonstrate the efficiency of the methods.