We propose the novel design, development, and implementation of the well-known generalized single step single solve (GS4) family of algorithms into the differential algebraic equation framework which not only allows altogether different numerical time integration algorithms within the GS4 family in each of the different subdomains but also additionally allows for the selection of different space discretized methods such as the finite element method and particle methods, and other spatial methods as well in a single analysis unlike existing state-of-the-art. For the first time, the user has the flexibility and robustness to embed different algorithms for time integration and different spatial methods for space discretization in a single analysis. In addition, the GS4 family enables a wide variety of choices of time integration methods in a single analysis and also ensures the second-order accuracy in time of all primary variables and Lagrange multipliers. This is not possible to date. However, the present framework provides the fusion of a wide variety of choices of time discretized methods and spatial methods and has the bandwidth and depth to engage in various types of research investigations as well and features for fine tuning of numerical simulations. It provides generality/versatility of the computational framework incorporating subdomains with different spatial and time integration algorithms with improved accuracy. The robustness and accuracy of the present work is not feasible in the current state of technology. Various numerical examples illustrate the significant capabilities and generality and effectiveness for general nonlinear dynamics.