### Abstract:

The 5-moment two-fluid plasma model uses Euler equations to describe the ion and electron fluids, and Maxwell's equations to describe the electric and magnetic fields. Two-fluid physics becomes significant when the characteristic spatial scales are on the order of the ion skin depth and characteristic time scales are on the order of the inverse ion cyclotron frequency. The two-fluid plasma model has disparate characteristic speeds ranging from the ion and electron speeds of sound to the speed of light. In addition, the characteristic frequencies in the system are the ion and electron plasma frequency, and the ion and electron cyclotron frequency. Explicit and implicit time-stepping schemes are explored for the two-fluid plasma model to study the accuracy and computational effectiveness with which they could capture two-fluid physics. The explicit schemes explored include the high resolution wave propagation method (a finite volume method) and the Runge-Kutta discontinuous Galerkin (RKDG) method (a finite element method). The ideal two-fluid model is a purely dispersive equation system with no physical or artificial dissipation. The dispersions are physical effects responsible for the wide variety of plasma waves; they are not numerical artifacts. This sets the two-fluid plasma model apart from other equation systems. The finite volume and finite element methods are compared for accuracy and computational expense for applications of the two-fluid plasma model. For realistic regimes, the explicit time-step for the two-fluid plasma model can be very restrictive making it computationally expensive. This motivates the implicit time-stepping scheme. A semi-implicit two-fluid plasma model is developed using the discontinuous Galerkin method where the electron fluid equations and Maxwell's equations are evolved implicitly eliminating the restrictions set by the speed of light, and the electron plasma and cyclotron frequencies. Resolving all ion time-scales is a minimum to capture two-fluid physics, so the ion fluid equations are solved explicitly. This allows for accuracy and physics considerations alone to determine the time-step. Non-ideal terms are added to the two-fluid plasma model in the form of resistivity, viscosity, and heat flux to provide a self-consistent and physically relevant two-fluid plasma model and these are compared to solutions of the ideal two-fluid plasma model. The two-fluid plasma model is compared to the more commonly used Hall-MHD model for accuracy and computational effort using an explicit time-stepping scheme. Simulations of two-fluid instabilities in the Z-pinch and the field-reversed configuration are presented in 3-dimensions.