High-Order Finite Element Method for High-Fidelity Plasma Modeling

Abstract

Plasmas can be described by a hierarchy of mathematical models: from the most general kinetic theory, which treats each constituent species as a continuous probability density function (PDF); to multi-fluid moment models that treat each species as a separate fluid; to single-fluid MHD models which introduce further simplifying assumptions such as charge neutrality. Moment models can assume small deviations away from thermodynamic equilibrium, e.g. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$5N$</tex> -moment multi-fluid plasma model, or can allow larger deviations, e.g. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$13N$</tex> -moment multi-fluid plasma model, which allows finite skew and kurtosis of the velocity PDF. Selecting a plasma model determines the degree of physical fidelity and the corresponding required computational effort. To enable self-consistent multiscale modeling of plasmas, this research develops the mathematical and numerical techniques to enable robust hybrid formulations <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> . The finite element method (FEM) is well suited for these continuum plasma models, which contain multiple spatial and temporal scales as well as contributions from hyperbolic, parabolic, and elliptic terms. Furthermore, the high-order-accurate FEM provides 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 multi-fluid plasma models are examples that benefit from high-order accuracy. The governing equations are expressed in a balance law form and solved using discontinuous Galerkin and hybridizable discontinuous Galerkin methods. The algorithms to solve the various plasma models are implemented in the flexible WARPXM code, which facilitates high-performance computing in an extensible framework. The computational methods are applied to investigate species separation in inertial confinement fusion capsules, electromagnetic wave propagation through plasma photonic crystals, disruption mechanisms of the plasma opening switch, drift wave turbulence in the Z pinch, and deviations away from thermodynamic equilibrium in collisionless magnetic reconnection.