A positivity-preserving discontinuous Galerkin (DG) scheme (Zhang, X. & Shu, C.W., J. Comp. Phys., 229(23), 8918-8934.) is used to solve the Extended Magnetohydrodynamics (XMHD) model, which is a two-fluid model expressed with a center-of-mass formulation. We prove that the DG scheme with a positivitypreserving limiter is stable for the system governed by the XMHD model or the resistive MHD model. We use the relaxation system formulation (Seyler, C.E., & Martin, M.R. Physics of Plasmas, 18, 012703.) for describing the XMHD model, and solve the equations using a split level implicit-explicit time advance scheme, stepping over the time step constraint imposed by the stiff source terms. The magnetic field is represented in an exact locally divergence-free form of DG (Li, F., & Shu, C.W. 22(1-3), 413-442.), which greatly improves the accuracy and stability of MHD simulations. As presently constructed, the method is able to handle a wide range of density variation, solve the XMHD model on MHD time scales, and provide greatly improved accuracy over a Finite Volume implementation of the same model. The extended-MHD code DG-PERSEUS, which is an implementation of this method on a 3D Cartesian coordinates, has been applied to the study of the magnetized shock in the context where a magnetized flow is interacting with a solid obstacle. Several physics issues are found to be associated with this problem, such as bow shock, reconnection, plasmoids, which have been studied. The inflow parameters, such as the magnetosonic mach number M f and the ratio of thermal pressure to magnetic pressure [beta] can significantly affect the physical structures of the flow-obstacle interaction, which can be used as a diagnostic tool for the flow. The Hall effect can also significantly influence the results. Interplanetary physics - the solar wind interacting with Mars - is also studied. Simulations are carried out to show that the interplanetary features (bow shock, reconnection) can also be achieved with laboratory parameters.