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Abstract
Plasma is the fourth state of matter, after solids, liquids, and gases; and furthermore, it is the most abundant form of ordinary matter in the universe largely due to the fact that stars are comprised of it. Plasma also occurs terrestrially in various engineering and physics applications, most notably in fusion reactors in the quest for humanity to create cleaner energy sources. Therefore, modeling and understanding plasma dynamics is of immense interest in the scientific community.
At very small scales, a plasma is a collection of electrically charged and neutral particles that interact through a variety of short, medium, and long scale forces. In a kinetic Vlasov description these particles are replaced by a probability distribution function that depends on time, space, and velocity to describe the distribution of particles, rather than tracking each individual particle. This reduction from particles to a statistical mechanics representation of the plasma massively reduces the degrees of freedom needed to describe the plasma. In particular, the starting point of the current work is the Vlasov-Poisson system, which is a system of integro-differential equations that describe the evolution of a collisionless plasma, under the additional assumption that particles only interact through electrostatic forces. However, even though the reduction from particles to distribution function greatly reduces the degrees of freedom, solving kinetic models such as Vlasov-Poisson is still prohibitively expensive on modern computers; this is due to the fact that the distribution function is still a function of seven independent variables (1 time, 3 space, 3 velocity).
In order to further reduce the complexity of the kinetic Vlasov description of plasma, one can employ a moment-closure approximation. The statistical moments of the distribution function (e.g., number density, mean velocity, standard deviation, etc.) are obtained by multiplying the distribution function by a polynomial in velocity coordinates, and then integrating over all of velocity space. Due to the velocity integration, the moments are by definition functions of only time and space, thus removing up to three independent variables, which greatly improves the computational expense of solving the evolution equations. However, solving directly for moments introduces a new challenge: the kinetic Vlasov system cannot be exactly reduced to a finite number of moments. That is, for every finite set of moments one may wish to evolve, there always exist higher-order moments that are needed in the evolution equations that are not in the finite set; this is known as the moment-closure problem.
Constructing suitable moment-closure approximations to kinetic models such as the Vlasov equations is a classical problem that dates back to the seminal work of Harald Grad in 1949 on Hermite polynomials. Since that work, many researchers have presented various solutions to the moment-closure problem, each with different approximation strategies and resulting properties. However, no moment-closure method exactly solves for the distribution function, and all approximations have challenges.
In the present work we introduce a novel moment-closure that we call the spectral element moment-closure. Standard moment-closures approximate the distribution function by a set of global moments. In this work, the idea is that we first divide the velocity space into a discrete mesh, and then introduce a finite set of moments that are local only to each velocity space element. In standard moment-closures the only mechanism to improve the approximation is to add more global moments; in the spectral element approach one can either vary the number of moments in each velocity element, vary the number of velocity elements, or both. An important advantage of this approach is that we are able to utilize in each velocity element a simple linear moment-closure that is provably symmetric hyperbolic, rather than a nonlinear closure that may be only conditionally hyperbolic.
Once the spectral element moment-closure has been established, this approach is then applied to solving the Vlasov-Poisson equation for plasma physics. Furthermore, a novel operator splitting method is developed, which allows the Vlasov-Poisson system to be handled in two distinct parts: (1) a series of decoupled local moment-closure problems; and (2) a series of decoupled equations that describe the information exchange between each of the local moment-closure problems. The resulting system of partial differential equations is implemented with a high-order discontinuous Galerkin method using Lax-Wendroff time-stepping. We verify the accuracy of the proposed scheme on a variety of test cases for the Vlasov-Poisson system, including a manufactured solution, weak and strong Landau damping, two-stream instability, and plasma sheath problem. We show that the total mass is exactly conserved by the numerical method, but that total momentum and total energy are not exactly conserved. However, in the numerical examples presented, we demonstrate that all of these quantities are very well-preserved even in complex numerical computations.