Higher-order generalized-α methods for hyperbolic problems

Our work addresses the hitherto-unfulfilled need for higher-order methods with dissipation control when applying highly-accurate and robust isogeometric analysis. The popular generalized-α time-marching method provides second-order accuracy in time and controls the numerical dissipation in the high-frequency regions of the discrete spectrum. It includes a wide range of time integrators as particular cases selected by appropriate parameters. Nevertheless, to exploit the spatial discretization's high-accuracy, in practice, we require high-order time marching methods that handle the poor approximability in the discrete high-frequency range. Thus, we extend the generalized-α method to increase its order of accuracy while keeping the unconditional stability behavior and the attractive user-control feature on the high-frequency numerical dissipation. A single parameter controls the dissipation, and the update procedure has the same structure as the original second-order method. That is, our high-order schemes require simple modifications of the available implementations of the generalized-α method.