The method of vertical lines provides a systematic instruction of the semi-discretization for space and time-dependent problems. The first step, namely the spatial discretization -- here applied to isothermal FE2 problems -- yields a huge system of differential-algebraic equations (DAE-system). The algebraic part stems from the equilibrium conditions and some constraint equations (in the case of displacement control, or for the transition of the deformation from macro to microscale), and the differential part is result of the evolution equations of the constitutive models on microscale (models of plasticity, viscoelasticity, or viscoplasticity). Frequently, the time discretization -- second step of the method of vertical lines -- of the resulting DAE-system is carried out using a Backward-Euler scheme. Here, however, we apply high-order stiffly-accurate, diagonally-implicit Runge-Kutta methods (SDIRK-methods) having the advantages that classical finite element implementations, which are based on a Backward-Euler implementation, imply a small modification, and offering the possibility to apply time-adaptive, i.e. step-size controlled computations.. The result of SDIRK-methods applied to DAE-systems is a system of non-linear equations in each stage. This system is very huge in the field of FE2-applications and particular approaches have to be investigated. A frequent approach to solve this system of non-linear equations is to apply the Multilevel-Newton algorithm, where the functional matrices are provided by numerical differentiation calling the finite element computations on microlevel. How the algorithmic structure looks like is not provided in the literature if no numerical differentiation is applied. This is demonstrated for the case of three-level Newton scheme which is based on the implicit function theorem. An alternative approach is related to the Newton-Schur scheme in a three-level version. Finally, a Chord-implementation is discussed so that the assembling of the tangents is minimized. An example is provided for relaxation dominated microstructures requiring time-adaptivity to estimate approriate step-sizes. Thus, the presentation provides both time-adaptive FE2 computations as well as theoretical aspects of the non-linear solution schemes.