Zap Energy is pursuing a sheared-flow-stabilized Z-pinch as the core of a fusion reactor technology. Recent experimental efforts have demonstrated currents higher than 600 kA and electron temperatures higher than lkeV. Some of the open questions for this concept are how the neutron yield scales with current, how the power supply bank couples to the dynamic plasma load, and how the cathodes survive under repeated discharges of several hundred klloJoules, among other questions. To help answer these questions, the Zap Energy Theory and Modeling team is using the WARPXM code, a discontinuous Galerkin modeling framework developed at U. Washington. Using a two-dimensional MHD model coupled to a circuit solver, we show scaling of neutron yields as a power law with current for several different voltage configurations, and discuss the implications for cathode erosion. We also discuss coupling of WARPXM with PIC models, benchmarking of WARPXM with NIMROD, and acceleration of WARPXM with GPU.

High-order accurate finite element methods provide unique benefits for problems that have strong anisotropies and complicated geometries and for stiff equation systems that are coupled through large source terms, e.g. Lorentz force, collisions, or atomic reactions. Magnetized plasma simulations of realistic devices using the kinetic or the multi-fluid plasma models are examples that benefit from highorder accuracy. The multi-fluid plasma model only assumes local thermodynamic equilibrium within each fluid, e.g. ion and electron fluids for the two-fluid plasma model. The algorithm 1 implements a discontinuous Galerkin method with an approximate Riemann solver to compute the fluxes of the fluids and electromagnetic fields at the computational cell interfaces. The multi-fluid plasma model has time scales on the order of the electron and ion cyclotron frequencies, the electron and ion plasma frequencies, the electron and ion sound speeds, and the speed of light. A general model for atomic reactions has been developed 2 and is incorporated in the multi-fluid plasma model. The multi-fluid plasma algorithm is implemented in a flexible code framework (WARPX) that allows easy extension of the physical model to include multiple fluids and additional physics. The code runs on multi-processor machines and is being adapted with OpenCL to many-core systems, characteristic of the next generation of high performance computers. The algorithm is applicable to study advanced physics calculations of plasma dynamics including magnetic plasma confinement and astrophysical plasmas. The discontinuous Galerkin method has also been applied to solve the Vlasov-Poisson kinetic model. Recently, a mixed finite element algorithm has been developed and implemented which exploits the expected physical behavior to apply either a discontinuous or continuous finite element representation, which improves computational efficiency without sacrificing accuracy.

Blended finite element (FE) methods combine high-order continuous and discontinuous spatial representations to exploit the expected physical behavior of the plasma dynamics. High-order accurate FE methods benefit problems that have strong anisotropies, complicated geometries, or stiff governing equations. The method is applicable to continuum models, such as continuum kinetic and multi-fluid plasma models. Discontinuous and blended FE methods are implemented in a flexible code framework, WARPX. The algorithm is applicable to study advanced physics calculations of plasma dynamics including HEDP, magnetic plasma confinement, and astrophysical plasmas. Owing to the complexity of plasma phenomena, a thorough understanding requires validated physical models, verified computational simulations, and well-diagnosed experiments. This paper focuses on developing computational methods for plasma models with sufficient physical and numerical fidelity to generate insight and predictability. Continuum Plasma Models Discrete models that account for each constituent particle is not particularly useful for the numerical treatment of realistic plasmas where the number of particles (N) and the number of interactions (> N2) is not computationally tractable. Instead an ensemble average is performed to give a statistical description. Plasmas may be most accurately modeled using kinetic theory, where distribution functions, fs(x,v), are governed by a Boltzmann equation ∂ fs ∂ t +v · ∂ fs ∂x + qs ms (E+v×B) · ∂ fs ∂v = ∂ fs ∂ t ∣∣∣∣ collisions (1) for each species s. Combined with Maxwell’s equations, the system leads to the continuum kinetic plasma model. Kinetic models in their most general form are six-dimensional, but reduced models, e.g. gyrokinetic, can also be meaningful. Further reduced plasma models result by taking moments over velocity space of Eq. (1) and of fs, the multi-fluid plasma model.[1] The principal variables of the multi-fluid plasma model are derived from moments of the distribution functions. The 5M model directly evolves the variables given by the first two moments and the tensor contraction of the third moment, ps = ρsTs = 1 3 ms ∫ w2 fs(v)dv, (2) where Ts is the temperature. The 13M model[2] directly evolves the variables given by the first 40 EPS Conference on Plasma Physics P2.409

High-order accurate finite element methods are important for problems that have strong anisotropies and complicated geometries and for stiff equation systems that are coupled through large source terms. Magnetized plasma simulations of realistic devices using the multi-fluid plasma model are examples that benefit from high-order accuracy. The multi-fluid plasma model only assumes local thermodynamic equilibrium within each fluid, e.g. ion and electron fluids for the two-fluid plasma model. Physical parameters indicate the importance of the two-fluid effects: electron to ion mass ratio, ion skin depth, and ion Larmor radius. The algorithm1 implements a discontinuous Galerkin method with an approximate Riemann solver to compute the fluxes of the fluids and electromagnetic fields at the computational cell interfaces. The multi-fluid plasma model has time scales on the order of the electron and ion cyclotron frequencies, the electron and ion plasma frequencies, the electron and ion sound speeds, and the speed of light. The multi-fluid plasma algorithm is implemented in a flexible code framework (WARPX) that allows easy extension of the physical model to include multiple fluids and additional physics. The code runs on multi-processor machines and is being adapted with OpenCL to many-core systems, characteristic of the next generation of high performance computers. WARPX has demonstrated a three-fluid (electrons, ions, and neutrals) simulation of a plasma sheath formation. Atomic reactions are incorporated that describe the effects of collisions between the species explicitly, allowing for the identification of regions of ionization/recombination, and interspecies momentum and energy transfer. The algorithm is validated with several test problems including the GEM challenge magnetic reconnection problem and the generation of dispersive plasma waves which are compared to analytical dispersion diagrams. The algorithm is applicable to study advanced physics calcu- ations of plasma dynamics including magnetic plasma confinement and astrophysical plasmas. Three-dimensional solutions of the Z-pinch and the field reversed configuration (FRC) magnetic plasma confinement configurations are presented.

The multi-fluid plasma model is derived from moments of the Boltzmann equation and typically has two fluids representing electron and ion species. Large mass differences between electrons and ions introduce disparate temporal and spatial scales and require a numerical algorithm with sufficient accuracy to capture the multiple scales. Source terms of the multi-fluid plasma model couple the fluids to themselves (interspecies interactions) and to the electromagnetic fields. The numerical algorithm must treat the inherent stiffness introduced by the multiple physical effects of the model and tightly couple the source terms of the governing equations. A discontinuous Galerkin method is implemented for the spatial representation. Time integration is investigated using explicit, implicit, semi-implicit methods. Semi-implicit treatment is accomplished using a physics-based splitting. The algorithm is applied to study drift turbulence in field reversed configuration plasmas to illustrate the physical accuracy of the model. The algorithm is also applied to plasma sheath formation which demonstrates Langmuir wave propagation.