ρ∞-Bathe method
What kind of problems is it mostly used for? Please describe.
The ρ∞-Bathe method (referred to here simply as the Bathe method) is mainly used for 2nd structural dynamics and wave propagation problems. It is intended to accurately resolve the important low frequency modes while providing controllable numerical dissipation of spurious high frequency modes
Describe the algorithm you’d like
Add the ρ∞-Bathe method alongside the existing Newmark/generalized-α family.
Unlike the previous method, Bathe is a composite implicit method that advances each timestep using two substeps. The original Bathe construction uses a trapezoidal rule first substep followed by a 3 point backward-Euler-type second substep. The ρ∞ extension introduces a parameter controlling the desired spectral radius at infinite frequency, providing tunable high frequency damping
Other implementations to know about
I am not aware of a general-purpose ODE implementation of the ρ∞-Bathe method. A somewhat related scheme is TRBDF2 due to the 2 step structure but I have not seen the Bathe method by itself.
References
ρ∞-Bathe method
What kind of problems is it mostly used for? Please describe.
The ρ∞-Bathe method (referred to here simply as the Bathe method) is mainly used for 2nd structural dynamics and wave propagation problems. It is intended to accurately resolve the important low frequency modes while providing controllable numerical dissipation of spurious high frequency modes
Describe the algorithm you’d like
Add the ρ∞-Bathe method alongside the existing Newmark/generalized-α family.
Unlike the previous method, Bathe is a composite implicit method that advances each timestep using two substeps. The original Bathe construction uses a trapezoidal rule first substep followed by a 3 point backward-Euler-type second substep. The ρ∞ extension introduces a parameter controlling the desired spectral radius at infinite frequency, providing tunable high frequency damping
Other implementations to know about
I am not aware of a general-purpose ODE implementation of the ρ∞-Bathe method. A somewhat related scheme is TRBDF2 due to the 2 step structure but I have not seen the Bathe method by itself.
References
G. Noh and K.-J. Bathe, "The Bathe time integration method with controllable spectral radius: The ρ∞-Bathe method," https://doi.org/10.1016/j.compstruc.2018.11.001
M. M. Malakiyeh, S. Shojaee, and K.-J. Bathe, "The Bathe time integration method revisited for prescribing desired numerical dissipation," DOI: https://doi.org/10.1016/j.compstruc.2018.10.008
G. Noh and K.-J. Bathe, "For direct time integrations: A comparison of the Newmark and ρ∞-Bathe schemes," DOI: https://doi.org/10.1016/j.compstruc.2019.05.01