Unconditionally stable higher-order Newmark methods by sub-stepping procedure Original Research Article

In this paper, a procedure to construct unconditionally stable higher-order step-by-step time integration algorithms based on the Newmark method is presented. The second-order accurate Newmark method is extended to third, fourth and even higher order by linearly combining the numerical results evaluated at some specific sub-step locations. The weighting factors and the sub-step locations are algorithmic parameters. They are chosen to eliminate the leading truncation error terms and to control the stability characteristics. The ranges of parameters for unconditionally stable third- and fourth-order algorithms are given. Besides, by using complex time steps, a third-orderL-stable and a fourth-order A-stable non-dissipative algorithms are derived. Furthermore, the accuracy of the excitation responses is shown to be improved by the present procedure as well. To maintain high-order accuracy, the excitation may need to be modified. The computational procedure is discussed. It is found that there are no changes in the implementation of the underlying Newmark method and no extra computations of high-order gradients. As the numerical results at the sub-step locations are obtained independent of one another, the evaluations can be computed in parallel. Comparisons with other higher-order algorithms are presented.