336 research outputs found
Accurate particle time integration for solving Vlasov-Fokker-Planck equations with specified electromagnetic fields
The Vlasov-Fokker-Planck equation (together with Maxwell's equations) provides the basis for plasma flow calculations. While the terms accounting for long range forces are established, different drift and diffusion terms are used to describe Coulomb collisions. Here, linear drift and a constant diffusion coefficient are considered and the electromagnetic fields are imposed, i.e., plasma frequency is not addressed. The solution algorithm is based on evolving computational particles of a large ensemble according to a Langevin equation, whereas the time step size is typically limited by plasma frequency, Coulomb collision frequency and cyclotron frequency. To overcome the latter two time step size constraints, a novel time integration scheme for the particle evolution is presented. It only requires that gradients of mean velocity, bath temperature, magnetic field and electric field have to be resolved along the trajectories. In fact, if these gradients are zero, then the new integration scheme is statistically exact; no matter how large the time step is chosen. Obviously, this is a computational advantage compared to classical integration schemes, which is demonstrated with numerical experiments of isolated charged particle trajectories under the influence of constant magnetic- and electric fields. Besides single ion trajectories, also plasma flow in spatially varying electromagnetic fields was investigated, that is, the influence of time step size and grid resolution on the final solution was studied
Moment-Based Accelerators for Kinetic Problems with Application to Inertial Confinement Fusion
In inertial confinement fusion (ICF), the kinetic ion and charge separation field effects may play a significant role in the difference between the measured neutron yield in experiments and the predicted yield from fluid codes. Two distinct of approaches exists in modeling plasma physics phenomena: fluid and kinetic approaches. While the fluid approach is computationally less expensive, robust closures are difficult to obtain for a wide separation in temperature and density. While the kinetic approach is a closed system, it resolves the full 6D phase space and classic explicit numerical schemes restrict both the spatial and time-step size to a point where the method becomes intractable. Classic implicit system require the storage and inversion of a very large linear system which also becomes intractable. This dissertation will develop a new implicit method based on an emerging moment-based accelerator which allows one to step over stiff kinetic time-scales. The new method converges the solution per time-step stably and efficiently compared to a standard Picard iteration. This new algorithm will be used to investigate mixing in Omega ICF fuel-pusher interface at early time of the implosion process, fully kinetically
A Deep Dive into the Distribution Function: Understanding Phase Space Dynamics with Continuum Vlasov-Maxwell Simulations
In collisionless and weakly collisional plasmas, the particle distribution function is a rich tapestry of the underlying physics. However, actually leveraging the particle distribution function to understand the dynamics of a weakly collisional plasma is challenging. The equation system of relevance, the Vlasov--Maxwell--Fokker--Planck (VM-FP) system of equations, is difficult to numerically integrate, and traditional methods such as the particle-in-cell method introduce counting noise into the distribution function.
In this thesis, we present a new algorithm for the discretization of VM-FP system of equations for the study of plasmas in the kinetic regime. Using the discontinuous Galerkin (DG) finite element method for the spatial discretization and a third order strong-stability preserving Runge--Kutta for the time discretization, we obtain an accurate solution for the plasma's distribution function in space and time.
We both prove the numerical method retains key physical properties of the VM-FP system, such as the conservation of energy and the second law of thermodynamics, and demonstrate these properties numerically. These results are contextualized in the history of the DG method. We discuss the importance of the algorithm being alias-free, a necessary condition for deriving stable DG schemes of kinetic equations so as to retain the implicit conservation relations embedded in the particle distribution function, and the computational favorable implementation using a modal, orthonormal basis in comparison to traditional DG methods applied in computational fluid dynamics.
A diverse array of simulations are performed which exploit the advantages of our approach over competing numerical methods. We demonstrate how the high fidelity representation of the distribution function, combined with novel diagnostics, permits detailed analysis of the energization mechanisms in fundamental plasma processes such as collisionless shocks. Likewise, we show the undesirable effect particle noise can have on both solution quality, and ease of analysis, with a study of kinetic instabilities with both our continuum VM-FP method and a particle-in-cell method.
Our VM-FP solver is implemented in the Gkyell framework, a modular framework for the solution to a variety of equation systems in plasma physics and fluid dynamics
A gyrokinetic model for the plasma periphery of tokamak devices
A gyrokinetic model is presented that can properly describe strong flows,
large and small amplitude electromagnetic fluctuations occurring on scale
lengths ranging from the electron Larmor radius to the equilibrium
perpendicular pressure gradient scale length, and large deviations from thermal
equilibrium. The formulation of the gyrokinetic model is based on a second
order description of the single charged particle dynamics, derived from Lie
perturbation theory, where the fast particle gyromotion is decoupled from the
slow drifts, assuming that the ratio of the ion sound Larmor radius to the
perpendicular equilibrium pressure scale length is small. The collective
behavior of the plasma is obtained by a gyrokinetic Boltzmann equation that
describes the evolution of the gyroaveraged distribution function and includes
a non-linear gyrokinetic Dougherty collision operator. The gyrokinetic model is
then developed into a set of coupled fluid equations referred to as the
gyrokinetic moment hierarchy. To obtain this hierarchy, the gyroaveraged
distribution function is expanded onto a velocity-space Hermite-Laguerre
polynomial basis and the gyrokinetic equation is projected onto the same basis,
obtaining the spatial and temporal evolution of the Hermite-Laguerre expansion
coefficients. The Hermite-Laguerre projection is performed accurately at
arbitrary perpendicular wavenumber values. Finally, the self-consistent
evolution of the electromagnetic fields is described by a set of gyrokinetic
Maxwell's equations derived from a variational principle, with the velocity
integrals of the gyroaveraged distribution function explicitly evaluated
High-order Discretization of a Gyrokinetic Vlasov Model in Edge Plasma Geometry
We present a high-order spatial discretization of a continuum gyrokinetic
Vlasov model in axisymmetric tokamak edge plasma geometries. Such models
describe the phase space advection of plasma species distribution functions in
the absence of collisions. The gyrokinetic model is posed in a four-dimensional
phase space, upon which a grid is imposed when discretized. To mitigate the
computational cost associated with high-dimensional grids, we employ a
high-order discretization to reduce the grid size needed to achieve a given
level of accuracy relative to lower-order methods. Strong anisotropy induced by
the magnetic field motivates the use of mapped coordinate grids aligned with
magnetic flux surfaces. The natural partitioning of the edge geometry by the
separatrix between the closed and open field line regions leads to the
consideration of multiple mapped blocks, in what is known as a mapped
multiblock (MMB) approach. We describe the specialization of a more general
formalism that we have developed for the construction of high-order,
finite-volume discretizations on MMB grids, yielding the accurate evaluation of
the gyrokinetic Vlasov operator, the metric factors resulting from the MMB
coordinate mappings, and the interaction of blocks at adjacent boundaries. Our
conservative formulation of the gyrokinetic Vlasov model incorporates the fact
that the phase space velocity has zero divergence, which must be preserved
discretely to avoid truncation error accumulation. We describe an approach for
the discrete evaluation of the gyrokinetic phase space velocity that preserves
the divergence-free property to machine precision
- …