Nonlinear Model Predictive Control (NMPC) and Moving Horizon Estimation (MHE) have become popular techniques for real-time control of various physical systems. However, in the context of embedded optimization, systems with fast dynamics still form a computational challenge. The simulation of the continuous time model and its sensitivity propagation are key tasks within any standard NMPC algorithm and have to be carried out frequently. We propose a general dynamic system structure that can handle all index-1 DAEs and enables us to exploit linear dependencies within the model. Additionally, we derive a tailored implicit Runge-Kutta (IRK) scheme that exploits this specific structure using a lifting-condensing approach and carrying out some computations offline. Moreover, we develop an algorithm to automatically transcribe continuous time models into this specific structure. The proposed method is applied to a wind turbine model, showing that the CPU time of the simulation and sensitivity propagation can be reduced by a factor greater than two.