A Discontinuous Galerkin Finite Element Method for a Class of Spherical Harmonic Expansions of the Boltzmann Equation

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.