D.E. Shumaker

SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers

SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.

Comparisons and physics basis of tokamak transport models and turbulence simulations

The predictions of gyrokinetic and gyrofluid simulations of ion-temperature-gradient (ITG) instability and turbulence in tokamak plasmas as well as some tokamak plasma thermal transport models, which have been widely used for predicting the performance of the proposed International Thermonuclear Experimental Reactor (ITER) tokamak [Plasma Physics and Controlled Nuclear Fusion Research, 1996 (International Atomic Energy Agency, Vienna, 1997), Vol. 1, p. 3], are compared. These comparisons provide information on effects of differences in the physics content of the various models and on the fusion-relevant figures of merit of plasma performance predicted by the models. Many of the comparisons are undertaken for a simplified plasma model and geometry which is an idealization of the plasma conditions and geometry in a Doublet III-D [Plasma Physics and Controlled Nuclear Fusion Research, 1986 (International Atomic Energy Agency, Vienna, 1987), Vol. 1, p. 159] high confinement (H-mode) experiment. Most of the mo...

Discrete Particle Noise in Particle-in-Cell Simulations of Plasma Microturbulence

Recent gyrokinetic simulations of electron temperature gradient (ETG) turbulence with flux-tube continuum codes vs. the global particle-in-cell (PIC) code GTC yielded different results despite similar plasma parameters. Differences between the simulations results were attributed to insufficient phase-space resolution and novel physics associated with toroidicity and/or global simulations. We have reproduced the results of the global PIC code using the flux-tube PIC code PG3EQ, thereby eliminating global effects as the cause of the discrepancy. We show that the late-time decay of ETG turbulence and the steady-state heat transport observed in these PIC simulations results from discrete particle noise. Discrete particle noise is a numerical artifact, so both these PG3EQ simulations and the previous GTC simulations have nothing to say about steady-state ETG turbulence and the associated anomalous heat transport. In the course of this work we develop three diagnostics which can help to determine if a particular PIC simulation has become dominated by discrete particle noise.

Simulation of ion temperature gradient turbulence in tokamaks

Results are presented from non-linear gyrokinetic simulations of toroidal ion temperature gradient turbulence and transport. The ion thermal fluxes are found to have an offset linear dependence on the temperature gradient and are significantly lower than gyrofluid or IFS-PPPL model predictions. A new phenomenon of non-linear effective critical gradients larger than the linear instability threshold gradients is observed and is associated with undamped flux surface averaged shear flows. The non-linear gyrokinetic codes have passed extensive tests, including comparison against independent linear calculations, a series of non-linear convergence tests and a comparison between two independent non-linear gyrokinetic codes. The most realistic simulations to date used actual reconstructed equilibria from experiments and a model for dilution by impurity and beam ions. These simulations highlight the importance of both self-generated and external E × B flow shear as well as the need for still more physics to be included.

Hybrid Ordered Particle Simulation (HOPS) code for plasma modelling on vector-serial, vector-parallel, and massively parallel computers

Abstract The Hybrid Ordered Particle Simulation (HOPS) code has been developed to provide a more efficient method for carrying out particle-in-cell (PIC) simulations of plasma phenomena. Conventional PIC methods store the particle attributes in tables that tend to have a random spatial order, which was appropriate for older serial-scalar computers. Problems associated with excessive accessing of memory, indirect indexing, and with many-to-one mappings in the deposition phase can make these codes inefficient on vector-serial, vector-parallel, and massively parallel machines. In the HOPS code we employ a sorting scheme to keep the particles ordered with respect to their spatial positions. By doing so, we have reduced memory accesses, recovered substantial direct indexing, and most importantly removed the many-to-one mapping problem. A low overhead sorting and reordering scheme is presented which allows HOPS to be most efficient on vector-serial machines and which scales linearly to various kinds of parallel computers. This paper focuses on the Cray C-90 vector-parallel computer but also discusses aspects of a massively parallel implementation.

Gyrokinetic simulations of ETG and ITG turbulence

We report on the resolution of a significant discrepancy between published continuum-code simulations and subsequent global particle-in-cell (PIC) simulations of electron-temperature-gradient (ETG) turbulence. Our investigations, using gyrokinetic δf -PIC- and continuum-code simulations and analytical theory, strongly support the conclusion from the earlier continuum-code simulations that ETG turbulence can drive the electron thermal conductivity χe large enough to be significant in some tokamaks. A successful ETG-turbulence benchmark between δf -PIC and continuum codes for ETG turbulence has also been completed. Scans in the magnetic shear show an abrupt transition to a high-χe state as the shear is increased from the benchmark value of s = 0. 1t o aboves = 0.4. When nonadiabatic ions are used, this abrupt transition is absent, and χe reaches values consistent with transport analyses of DIII-D, JET, JT60-U and NSTX discharges. The balances of zonal-flow driving and damping terms in late-time quasi-steady phase of ITG turbulence have been unfolded using a new run-time gyrokinetic-simulation diagnostic. The zonal flow level is set by a balance of large driving and damping terms which almost cancel each other. The driving is found to be mostly by the Reynolds stress, while the dissipation is mostly by the linear (transittime) damping terms. It is also shown that useful zonal-flow-balance information can be obtained with spatially localized samples at as few as four poloidal locations. Real-geometry simulations have been undertaken, using the nonlinear δf -PIC gyrokinetic code SUMMIT/PG3EQ NC, of the DIII-D ‘Cyclone’ shot #81499 and of shot #118561, which had broad-wavenumber-range density fluctuation measurements. Real geometry is found to have a significant effect on the transport rates, even though the effect on the linear growth rates is often modest.

Interactive Scientific Exploration of Gyrofluid Tokamak Turbulence

This research is a part of the Numerical Tokamak Project, a national consortium of efforts to create predictive nu merical simulations of fluid plasma turbulence in tokamak fusion experiments using the most powerful supercomput ers in the world. Major progress has been made in fusion research, as demonstrated by the recent production of 10 megawatts of fusion power in the Princeton Tokamak Fusion Test Reactor (TFTR). However, much research is still needed before fusion can be a commercially success ful electricity source. High-performance computing will play a profoundly important role in designing these ma chines. Realistic simulations are an important component of this research. The mission of this work is to use mas sively parallel computers to simulate the behavior of tokamak plasma and represent the results of the simula tion for scientific visualization and diagnosis. These simu lations have begun to produce results that are encourag ingly close to present experiments. As this trend continues and as current models evolve, the simulations will provide an increasingly valuable tool for optimizing the design of future tokamaks, potentially reducing their cost and in creasing the certainty of meeting their objectives.

High-performance climate system modeling using a domain and task decomposition message-passing approach

We have developed a climate system modeling framework (CSMF) for high-performance systems, designed to schedule and couple multiple physics simulation packages in a flexible and transportable manner. Some of the major packages in the CSMF include models of atmospheric and oceanic circulation and chemistry, land surface and sea ice processes, and trace gas biogeochemistry. Parallelism is achieved through both domain decomposition and process-level concurrency, with data transfer and synchronization accomplished through message-passing. Both machine transportability and architecture-dependent optimization are handled through libraries and conditional compile directives. Preliminary experiments with the CSMF have been executed on a number of high-performance platforms, including the Intel Paragon, the TMC CM-5 and the Meiko CS-2, and we are in the very early stages of optimization. Progress to date (1994) is presented.<<ETX>>