As gravitational wave astronomy begins to mature, the need for precise and efficient numerical modelling of compact object mergers grows. Simulations of mergers require relativistic models capable of capturing complex dynamics that act over disparate length and time scales, where improving the quality of the solution always comes at the cost of additional computation. In the vast majority of merger simulations, the equations defining the interaction of the matter and electromagnetic fields make a number of specific assumptions. In this work, we provide several means of efficiently evolving systems of equations that move away from this ideal framework. So-called non-ideal models of magnetohydrodynamics (MHD) may be dissipative or describe the evolution of multiple fluid species. Here, we present both kinds, describe why they are more physically accurate than the models that are generally used for merger simulations, and demonstrate the drawbacks inherent in their use. To alleviate the computational cost incurred by some non-ideal models, we develop a number of methods. First, we demonstrate how resistive and multiple fluid models of MHD can be accelerated by 21× by executing the most computationally expensive tasks on graphics processing units. We then focus our attention on resistive MHD, developing an approximation that can extend the ideal models that are currently used, emulating a full resistive model within a fraction of the time. After modifying this approximation to work in general relativity, we then use it for resistive simulations of neutron star mergers. These simulations demonstrate conclusively that a physically realistic magnitude of resistivity for neutron star matter can have dramatic effects upon the evolution of merger simulations, with large impacts on key observables. Finally, we present two novel, hybrid models of MHD that allow for significantly improved accuracy and computational efficiency when evolving resistive MHD. With all these tools, the computational costs preventing the widespread adoption of non-ideal models for merger simulations have been dramatically reduced, allowing for higher resolution and more realistic, resistive descriptions of neutron star matter for future merger simulations.