High-order implicit time integration scheme with controllable numerical dissipation based on mixed-order Padé expansions

A single-step high-order implicit time integration scheme with controllable numerical dissipation at high frequency is presented for the transient analysis of structural dynamic problems. The amount of numerical dissipation is controlled by a user-specified value of the spectral radius $\rho_\infty$ in the high frequency limit. Using this user-specified parameter as a weight factor, a Pad\'e expansion of the matrix exponential solution of the equation of motion is constructed by mixing the diagonal and sub-diagonal expansions. An efficient time stepping scheme is designed where systems of equations similar in complexity to the standard Newmark method are solved recursively. It is shown that the proposed high-order scheme achieves high-frequency dissipation while minimizing low-frequency dissipation and period errors. The effectiveness of dissipation control and efficiency of the scheme are demonstrated with numerical examples. A simple recommendation on the choice of the controlling parameter and time step size is provided. The source code written in MATLAB and FORTRAN is available for download at: https://github.com/ChongminSong/HighOrderTimeIntegration.