For direct time integrations: A comparison of the Newmark and ρ_∞-Bathe schemes

We consider the unconditionally stable Newmark and ρ_∞-Bathe methods for the direct time integration of the finite element equations in structural dynamics and wave propagations. In our evaluation of the Newmark method we consider the parameters δ and α, and in the ρ_∞-Bathe method we consider the parameters γ and ρ_∞, with 0<γ<∞,γ≠1 and ρ_∞∈[-1,+1]. We show that the Newmark method as usually used with its δ and α parameters, α=0.25(δ+0.5)² and δ⩾0.5, is a special case of the ρ_∞-Bathe method. We also show that the β₁/β₂-Bathe method is a special case of the ρ_∞-Bathe scheme. The study of the curves of numerical dissipation and dispersion shows that the ρ_∞-Bathe method provides effective dissipation and dispersion whereas the Newmark method lacks in that regard. To illustrate our theoretical findings we give the results of some example solutions of structural dynamics and wave propagations. Our study also shows that further research is needed to identify the optimal use of the ρ_∞-Bathe scheme and other implicit methods in wave propagation analyses.