Algorithm for time-delay interferometry numerical simulation and sensitivity investigation

In this work, we introduce a generic algorithm to numerically determine the time delays and spacecraft positions for a time-delay interferometry (TDI) channel in the dynamical case, and streamline the calculations by implementing an SC layout-time delay diagram. We select 11 second-generation TDI channels constructed from four approaches and evaluate their performances including gravitational wave responses, noise levels, and averaged sensitivities under a numerical LISA orbit. The results show that the interference paths of selected TDI channels are well matched and the laser frequency noise should be suppressed under the secondary noise. The channels show various sensitivities in the range of [0.1 mHz, 0.3 Hz], and the major differences appear in the frequency region lower than 20 mHz. The optimal channel A$_2$ or E$_2$ combined from second-generation Michelson TDI channels (X$_1$, X$_2$, and X$_3$) achieves the best sensitivity in the selected channels for the frequency lower than 50 mHz, while the Sagnac $\alpha_1$ channel shows the worse sensitivity. Multiple channels show better sensitivities at some characteristic frequencies compared to the fiducial X$_1$ channel. The Michelson-type channels would have identical sensitivities considering noise level changes with the GW response. The joint $\mathrm{A_2+E_2+T_2}$ observation not only enhances the sensitivity of the X$_1$ channel by a factor of $\sqrt{2}$ to 2 but also improves the capacity of sky coverage.