This master's thesis explores the application of a matrix-free GMRES solver for implicit time integration of unsteady turbulent flows using a high-order DGSEM spatial discretisation. Efficient time-integration methods remain one of the primary challenges towards enabling industrial use of high-order methods for scale-resolving simulations. The focus of this study is on the development of such a prototype implicit solver for the DGSEM solver in the CFD framework TRACE, developed by DLR, which specialises in internal aerodynamics of turbomachinery components. To gain a fundamental understanding of the method and have a lightweight experimentation tool, a one-dimensional DGSEM solver that solves the Burgers equation was first developed from scratch. With this gained knowledge, the implementation of the solver in TRACE could start. The prototype solver developed in TRACE is capable of using both finite-difference and forward automatic differentiation techniques to enable the matrix-free GMRES method. Performance tests show that the automatic differentiation method is approximately twice as slow as the finite-difference differentiation method when comparing them for an equal number of linear iterations. However, the finite-difference method produces inaccurate gradient estimations that could influence the linear solve. Lastly, the solver was applied to perform an LES of a turbulent channel flow. The implicit solver yielded similar results compared to the time-explicit reference simulations at time step sizes greater than the explicit limit. Using the finite-difference differentiation method, the implicit solver was also found to require fewer CPU hours than the explicit reference simulations.