Daniel C. Barnes

Semi-implicit magnetohydrodynamic calculations

Abstract A semi-implicit algorithm for the solution of the nonlinear, three-dimensional, resistive MHD equations in cylindrical geometry is presented. The specific model assumes uniform density and pressure, although this is not a restriction of the method. The spatial approximation employs finite differences in the radial coordinate, and the pseudo-spectral algorithm in the periodic poloidal and axial coordinates. A leapfrog algorithm is used to advance wave-like terms; advective terms are treated with a simple predictor-corrector method. The semi-implicit term is introduced as a simple modification to the momentum equation. Dissipation is treated implicitly. The resulting algorithm is unconditionally stable with respect to normal modes. A general discussion of the semi-implicit method is given, and specific forms of the semi-implicit operator are compared in physically relevant test cases. Long-time simulations are presented.

An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm

Abstract This paper discusses a novel fully implicit formulation for a one-dimensional electrostatic particle-in-cell (PIC) plasma simulation approach. Unlike earlier implicit electrostatic PIC approaches (which are based on a linearized Vlasov–Poisson formulation), ours is based on a nonlinearly converged Vlasov–Ampere (VA) model. By iterating particles and fields to a tight nonlinear convergence tolerance, the approach features superior stability and accuracy properties, avoiding most of the accuracy pitfalls in earlier implicit PIC implementations. In particular, the formulation is stable against temporal (Courant–Friedrichs–Lewy) and spatial (aliasing) instabilities. It is charge- and energy-conserving to numerical round-off for arbitrary implicit time steps (unlike the earlier “energy-conserving” explicit PIC formulation, which only conserves energy in the limit of arbitrarily small time steps). While momentum is not exactly conserved, errors are kept small by an adaptive particle sub-stepping orbit integrator, which is instrumental to prevent particle tunneling (a deleterious effect for long-term accuracy). The VA model is orbit-averaged along particle orbits to enforce an energy conservation theorem with particle sub-stepping. As a result, very large time steps, constrained only by the dynamical time scale of interest, are possible without accuracy loss. Algorithmically, the approach features a Jacobian-free Newton–Krylov solver. A main development in this study is the nonlinear elimination of the new-time particle variables (positions and velocities). Such nonlinear elimination, which we term particle enslavement, results in a nonlinear formulation with memory requirements comparable to those of a fluid computation, and affords us substantial freedom in regards to the particle orbit integrator. Numerical examples are presented that demonstrate the advertised properties of the scheme. In particular, long-time ion acoustic wave simulations show that numerical accuracy does not degrade even with very large implicit time steps, and that significant CPU gains are possible.

Kinetic tilting stability of field-reversed configurations

Stability of the internal tilting mode in an elongated prolate field‐reversed configuration (FRC) is investigated numerically. Eigenfrequencies are calculated from a Vlasov‐fluid dispersion functional that is separated into fluid and kinetic portions. The latter are evaluated by a Monte Carlo method following a sample of the equilibrium orbits. An innovative Fourier transform technique is developed to reduce the operation count of the algorithm. Kinetic growth rates obtained for an experimentally relevant equilibrium indicate essential stabilization of internal tilting in an FRC with s≲2, where s measures the number of thermal gyroradii in the configuration. For s→∞ the kinetic growth rates approach previous magnetohydrodynamic (MHD) values.

Review of the Los Alamos FRX-C experiment

The FRX-C device is a large field-reversed theta pinch experiment with linear dimensions twice those of its FRX-A and FRX-B predecessors. It is used to form field-reversed configurations (FRCs), which are high-beta, highly prolate compact toroids. The FRX-C has demonstrated an R/sup 2/ scaling for particle confinement in FRCs, indicating particles are lost by diffusive processes. Particle losses were also observed to dominate the energy balance. When weak quadrupole fields were applied to stabilize the n = 2 rotational mode, FRC lifetimes >300..mu..s were observed. Detailed studies of the FRC equilibrium were performed using multichord and holographic interferometry. Measurements of electron temperature by Thomson scattering showed a flat profile and substantial losses through the electron channel. The loss rate of the internal poloidal flux of the FRC was observed to be anomalous and to scale less strongly with temperature than predicted from classical resistivity.

An efficient mixed-precision, hybrid CPU-GPU implementation of a nonlinearly implicit one-dimensional particle-in-cell algorithm

Recently, an implicit, nonlinearly consistent, energy- and charge-conserving one-dimensional (1D) particle-in-cell method has been proposed for multi-scale, full-f kinetic simulations [G. Chen et al., J. Comput. Phys. 230 (18) (2011)]. The method employs a Jacobian-free Newton-Krylov (JFNK) solver, capable of using very large timesteps without loss of numerical stability or accuracy. A fundamental feature of the method is the segregation of particle-orbit computations from the field solver, while remaining fully self-consistent. This paper describes a very efficient, mixed-precision hybrid CPU-GPU implementation of the 1D implicit PIC algorithm exploiting this feature. The JFNK solver is kept on the CPU in double precision (DP), while the implicit, charge-conserving, and adaptive particle mover is implemented on a GPU (graphics processing unit) using CUDA in single-precision (SP). Performance-oriented optimizations are introduced with the aid of the roofline model. The implicit particle mover algorithm is shown to achieve up to 400GOp/s on a Nvidia GeForce GTX580. This corresponds to 25% absolute GPU efficiency against the peak theoretical performance, and is about 100 times faster than an equivalent single-core CPU (Intel Xeon X5460) compiler-optimized execution. For the test case chosen, the mixed-precision hybrid CPU-GPU solver is shown to over-perform the DP CPU-only serial version by a factor of ~100, without apparent loss of robustness or accuracy in a challenging long-timescale ion acoustic wave simulation.

Short Note: A charge- and energy-conserving implicit, electrostatic particle-in-cell algorithm on mapped computational meshes

We describe the extension of the recent charge- and energy-conserving one-dimensional electrostatic particle-in-cell algorithm in Ref. [G. Chen, L. Chacon, D.C. Barnes, An energy- and charge-conserving, implicit electrostatic particle-in-cell algorithm, Journal of Computational Physics 230 (2011) 7018-7036] to mapped (body-fitted) computational meshes. The approach maintains exact charge and energy conservation properties. Key to the algorithm is a hybrid push, where particle positions are updated in logical space, while velocities are updated in physical space. The effectiveness of the approach is demonstrated with a challenging numerical test case, the ion acoustic shock wave. The generalization of the approach to multiple dimensions is outlined.

The Liouville theorem and accurate plasma simulation

Abstract The cold two-stream instability has been examined within the context of the one-dimensional Vlasov-Poisson model for three simulation methods. Two of the methods treat a continuum of particles; the third is the particle-in-cell method. The onset of noise in the simulations occurred near the time when the beams crossed or nearly crossed in the phase space, indicating a violation of Liousvilles theorem or strong sensitivity of particle trajectories to initial conditions in the vicinity of the point of near crossing. Details of the underlying Hamiltonian structure of the particle dynamics appear to be important in achieving accurate plasma simulations.

The dispersion functional for multidimensional equilibria

Numerical study of the linear stability of plasmas is very difficult when one or more of the plasma species is collisionless and the equilibrium is multidimensional, that is, characterized by two or more nonignorable spatial coordinates. The problem arises, for example, in evaluating kinetic stabilizing effects on the internal tilting mode (an n=1 ballooning mode) in field‐reversed configurations. In this paper, the Laplace transform of the perturbation distribution function for a collisionless species is derived for all classes of phase‐space trajectories and used to construct the dispersion functional for multidimensional equilibria. The kinetic part of the dispersion functional is expressed in terms of the Laplace transform of autocorrelation functions with respect to a certain delay time. It is shown how to obtain the same result formally by using Liouville eigenfunctions. For the case of the Vlasov‐fluid model, the dispersion functional is transformed in a way that is particularly appropriate for computation of the kinetic stability of field‐reversed configurations to the internal tilting mode.

Magnetohydrodynamic simulation of coronal magnetic fields

Abstract The application of supercomputers and advanced numerical techniques to problems of coronal structure and dynamics is described. Numerical methods appropriate for the long time scale simulation of nonlinear magnetohydrodynamic systems are discussed. Three specific examples of the application of these techniques to the solar corona are given. These are magnetic energy storage and conversion, a model for steady coronal heating, and calculation of stable force-free equilibria from given boundary data, such as that obtained with a vector magnetograph. It is suggested that the continued application of these methods will result in substantial advances in the understanding of coronal dynamics and structure.

Mechanical injection of magnetic helicity

A novel technique for the injection of magnetic helicity into a closed volume is described. In this new approach, mechanical energy is converted directly into the production of magnetic helicity. Several geometries illustrate the flexibility of the approach. The application of mechanical injection of magnetic helicity to a magnetically insulated impact fusion system is briefly described. In this application, a β≫1 spheromak plasma is produced by impact of electrically conducting shells. Magnetic helicity is injected by the relative motion of these shells and subsequent plasma relaxation produces the spheromak magnetic field.