Higher order time-step integration methods with complex time steps

Unconditionally stable higher order time-step integration algorithms are presented. The algorithms are based on the Newmark method with complex time steps. The numerical results at the (complex) sub-step locations are combined linearly to give higher order accurate results at the end of the time step. The ultimate spectral radius in the high-frequency range is a controllable parameter for these algorithms. Among these algorithms, the asymptotic annihilating algorithm and the non-dissipative algorithm correspond to the first sub-diagonal and diagonal Pade approximations respectively. The characteristics of the present algorithms with various numerical dissipations are found to be in between these two algorithms. The algorithmic parameters for the third, fifth and seventh order algorithms with various numerical dissipations are given explicitly. The order of accuracy is increased by one if these algorithms are set to non-dissipative. The spectral radii, algorithm damping ratios and relative period errors are compared favourably with other higher order algorithms. (C) 1998 Academic Press Limited.