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The kinetic theory of plasmas is governed by the Boltzmann equation, which describes the evolution of the distributions of electrons and ion species within six-dimensional phase space. When equipped with an appropriate model for the various binary collision events that can occur, this equation can be used to simulate a wide variety of plasma physics problems. Numerical solutions of the Boltzmann equation can be generated by either directly discretizing the six-dimensional phase space or by utilizing particle-based techniques, such as the Particle-in-Cell (PIC) method, to describe the statistical evolution of the distribution functions. However, both methods are inherently computationally expensive, and solutions in complex three-dimensional spatial domains remain intractable. For highly collisional plasmas in which the velocity distributions are nearly isotropic, a spherical harmonic expansion can be used to reduce the dimensionality of the phase space. Recent work at the Naval Research Laboratory 1 has re-examined this classical analytical approach in the context of low-temperature collisional plasmas. The authors derived a multi-term expansion that reduces the six-dimensional Boltzmann equation to a set as four-dimensional fluid-like equations. In this work, we describe the development of a general, high-order numerical solution method for this set of equations based on the discontinuous Galerkin (DG) finite element method. The electron velocity distribution function is discretized directly on the 4D phase-space mesh. Self-consistent electric and magnetic fields are simulated by solving Maxwell’s equations on a sub-dimensional manifold within the higher-dimensional phase-space mesh. A description of the numerical methodology will be presented along with a series of test problems that provide a demonstration of the algorithm and an initial examination of the benefits associated with using high-order phase-space discretizations. |
