An analysis of implicit time integration schemes for wave propagations. (1st April 2020) - Kwon, Sun-Beom; Bathe, Klaus-Jürgen; Noh, Gunwoo
Highlights: We consider the Bathe time integration schemes for wave propagation problems. New insights into the standard, β 1 / β 2 - and ρ ∞ -Bathe methods are presented. We show that the ρ ∞ -Bathe scheme may provide similar results as the standard Bathe scheme, but with larger time step size. Some example solutions of wave propagations are presented to illustrate the theoretical findings. Abstract: The objective of this paper is to investigate the optimal use of some time integration schemes for the solution of transient wave propagation problems. We study the accuracy characteristics of the trapezoidal rule and the ρ ∞ -Bathe scheme considering various parameter sets ( ρ ∞, γ, CFL ) with both consistent and lumped mass matrices. The ρ ∞ -Bathe scheme includes also the standard-, β 1 / β 2 -Bathe methods, the Newmark method and the trapezoidal rule. The study of the numerical dispersion shows that in the case of the consistent mass matrix, the ρ ∞ -Bathe scheme with a proper setting of ( ρ ∞, γ ) and standard Bathe scheme provide similar dispersion errors and outperform the trapezoidal rule. The optimal CFL number of the ρ ∞ -Bathe scheme is about 25% larger than for the standard Bathe scheme. In addition, we show that using a lumped mass matrix and proper values of ρ ∞ < 0, γ and CFL in the ρ ∞ -Bathe scheme, more accurate solutions can be obtained in some analyses.