The Bathe time integration method with controllable spectral radius: The ρ∞-Bathe method

•We focus on the Bathe time integration for dynamic solutions.•The time step splitting ratio and spectral radius are considered as two parameters.•The aim is to prescribe the amplitude decay (dissipation) in an optimum manner.•We show that the two parameters are effective and illustrate their use. We consider the Bathe implicit time integration method and focus on the time step splitting ratio and the spectral radius at large time steps to improve and generalize the scheme. The objective is to be able to prescribe the amplitude decay (dissipation) and period elongation (dispersion) for the numerical integration, and to achieve this aim in a direct and optimum manner with the minimum number of parameters. We show that the use of the time step splitting ratio and spectral radius is effective to prescribe in a smooth manner no amplitude decay to very large amplitude decays, with correspondingly small period elongation to very large period elongations while maintaining second-order accuracy. We analyze the effects of the splitting ratio and spectral radius on the stability and accuracy of the scheme and illustrate the use of these parameters in comparison with previously published methods. Furthermore, we show that with a proper setting of these parameters more accurate results may be obtained in some analyses.