Unconditionally stable higher-order Newmark methods by sub-stepping procedure (Q1376128)

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 orders 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-order \(L\)-stable and a fourth-order \(A\)-stable non-dissipative algorithm are derived.