Operator Split (OS), Picard iteration and Jacobian-free Newton Krylov (JFNK) methods based on nonlinear coarse mesh finite difference (CMFD) with nodal expansion methods (NEM) are respectively introduced to solve the transient full core coupled models in the COupling Multiphysics Environment (COME). In the COME, in addition to the traditional OS and Picard coupled methods, we develop two different JFNK-based coupled methods with physics and algebraic-based hybrid preconditioners at each time step: OS_JFNK and JFNK coupled methods. Specifically, the efficient JFNK method is applied to solve CMFD discrete models of only neutronics in the OS framework for OS_JFNK coupled strategies or whole Neutronics-Thermal Hydraulic (N-TH) coupled models for JFNK coupled methods. Then nonlinear corrective coupling coefficients are updated by the two-node NEM every few Newton steps at each time step. Six cases of NEACRP 3D core transient benchmarks are analyzed and numerical solutions of OS, OS_JFNK, Picard and JFNK methods in the COME generally agree well with other published solutions. JFNK and Picard coupled methods can track the reference solutions better than OS and OS_JFNK coupled methods due to multiple iterations or simultaneous solutions between neutronics and TH models in the tight coupled form. Furthermore, OS_JFNK or JFNK methods can obtain higher computational efficiency than OS or Picard methods with or without the fission source extrapolation acceleration, respectively, which indicates the potential of the developed COME using different coupled methods for the transient full core multiphysics coupled problems.