Publications¶
In the following list, a conference proceedings paper is included only if the paper was peer-reviewed.
17. A. H. Hakim. G.W Hammett and E. L. Shi, “On discontinuous Galerkin discretizations of second-order derivatives”, Journal of Computational Physics, Submitted (2014)
Abstract Some properties of a Local discontinuous Galerkin (LDG) algorithm are demonstrated for the problem of evaluting a second derivative \(g = f_{xx}\) for a given \(f\). (This is a somewhat unusual problem, but it is useful for understanding the initial transient response of an algorithm for diffusion equations.) LDG uses an auxiliary variable to break this up into two first order equations and then applies techniques by analogy to DG algorithms for advection algorithms. This introduces an asymmetry into the solution that depends on the choice of upwind directions for these two first order equations. When using piecewise linear basis functions, this LDG solution \(g_h\) is shown not to converge in an \(L_2\) norm because the slopes in each cell diverge. However, when LDG is used in a time-dependent diffusion problem, this error in the second derivative term is transient and rapidly decays away, so that the overall error is bounded. I.e., the LDG approximation \(f_h(x,t)\) for a diffusion equation \(\partial f / \partial t = f_{xx}\) converges to the proper solution (as has been shown before), even though the initial rate of change \(\partial f_h / \partial t\) does not converge. We also show results from the Recovery discontinuous Galerkin (RDG) approach, which gives symmetric solutions that can have higher rates of convergence for a stencil that couples the same number of cells.
16. Thomas G Jenkins, Travis M Austin, David N Smithe, John Loverich, and Ammar H Hakim, “Time-domain simulation of nonlinear radiofrequency phenomena”, Physics of Plasmas, 20, 012116 (2013)
Abstract Nonlinear effects associated with the physics of radiofrequency wave propagation through a plasma are investigated numerically in the time domain, using both fluid and particle-in-cell (PIC) methods. We find favorable comparisons between parametric decay instability scenarios observed on the Alcator C-MOD experiment [J. C. Rost, M. Porkolab, and R. L. Boivin, Phys. Plasmas 9, 1262 (2002)] and PIC models. The capability of fluid models to capture important nonlinear effects characteristic of wave-plasma interaction (frequency doubling, cyclotron resonant absorption) is also demonstrated.
15. A. H. Hakim, T.D. Rognlien, R.J. Groebner, J. Carlsson, J.R. Cary, S.E. Kruger, M. Miah, A. Pankin, A. Pletzer, S. Shasharina, S. Vadlamani, R. Cohen and T. Epperly. “Coupled Core-Edge Simulations of H-Mode Buildup Using the Fuson Application for Core-Edge Transport Simulations (FACETS) Code”, Physics of Plasmas, 19, 032505 (2012)
Abstract Coupled simulations of core and edge transport in the DIII-D shot number 118897, after the L-H transition but before the first edge localized mode (ELM), are presented. For the plasma core transport, a set of one dimensional transport equations are solved using the FACETS:Core solver. The fluxes in this region are calculated using the GLF23 anomalous transport model and Chang- Hinton neoclassical model. For the plasma edge transport, two-dimensional transport equations are solved using the UEDGE code. Fluxes in the edge region use static diffusivity profiles based on an interpretive analysis of the experimental profiles. Simulations are used to study the range of validity of the selected models and sensitivity to neutral fueling. It has been demonstrated that the increase of neutral influx to the level that exceeds the level of neutral influx obtained from analysis simulations with the UEDGE code by a factor of two results in increased plasma density pedestal heights and plasma density levels in the scrape-off-layer region. However, the additional neutral influx has relatively weak effect on the pedestal width and plasma density profiles in the plasma core for the DIII-D discharge studied in this research.
14. B. Srinivasan, A. Hakim and U. Shumlak. “Numerical Methods for Two-Fluid Dispersive Fast MHD Phenomena”, Communications in Computational Physics, 10, Pg. 183-215, 2011.
Abstract The high-resolution wave propagation method and the Runge-Kutta discontinuous Galerkin (RKDG) method are studied for applications to balance laws describing plasma fluids. The plasma fluid equations explored are dispersive and not dissipative. The physical dispersion introduced through the source terms leads to the wide variety of plasma waves. The dispersive nature of the plasma fluid equations explored separates the work in this paper from previous publications. The linearized Euler equations with dispersive source terms are used as a model equation system to compare the wave propagation and RKDG methods. The numerical methods are then studied for applications of the full two-fluid plasma equations. The two-fluid equations describe the self-consistent evolution of electron and ion fluids in the presence of electromagnetic fields. It is found that the wave propagation method, when run at a CFL number of 1, is more accurate for equation systems that do not have disparate characteristic speeds. However, if the oscillation frequency is large compared to the frequency of information propagation, source splitting in the wave propagation method may cause phase errors. The Runge-Kutta discontinuous Galerkin method provides more accurate results for problems near steady-state as well as problems with disparate characteristic speeds when using higher spatial orders.
13. John Loverich, Ammar Hakim, Sudhakar Mahalingam, Peter Stoltz, Sean C.D. Zhou, Michael Keidar, M. Jandrapu, TaiSen Zhuang, Jason Cassibry and Richard Hatcher. “Simulation of labratory accretion disk and weakly ionized hypersonic flows using Nautilus”, American Institute of Aeronautics and Astronautics. Paper number AIAA 2011-4012, 2011.
Abstract A proposed experiment at HyperV technologies in partnership with Los Alamos National Lab will explore the use of merging, high velcocity (50km/s) plasma jets as a laboratory model for astrophysical accretion disks. Similarly, researchers at George Washington Uni- versity (GWU) are currently developing an experiment for investigating weakly ionized hypersonic flow in magnetic fields. The two experiments cover a broad range of hypersonic conducting fluid flows which can be explored using the shock capturing plasma fluid code, Nautilus. In this paper, progress on modeling both experiments is presented along with associated physics and numerical techniques.
12. Ammar H. Hakim and John Loverich. “Robust, positivity and divergence preserving code for multi-fluid, multi-species electromagnetics and plasma physics applications”, American Institute of Aeronautics and Astronautics. Paper number AIAA 2011-4011, 2011.
Abstract In this paper we will present algorithms implemented in Nautilus, a new code for the solution of a wide variety of plasma and neutral fluid equations. Nautilus can simulate fully and weakly ionized flows and has been applied to fast magnetohydrodynamics (MHD) devices like Z-pinches and dense-plasma focuses, chemically reacting hypersonic flows and plasma jet propagation and merging for the creation of high energy-density states in the laboratory. We present details of new positivity preserving fluid algorithms as well as collocated Maxwell equation solvers that preserve divergence and work on general geometries. Current state of Nautilus is described and future directions outlined.
11. John Loverich, Ammar Hakim and Uri Shumlak. “A Discontinuous Galerkin Method for Ideal Two-Fluid Plasma Equations”, Communications in Computational Physics, 9 (2), Pg. 240-268, 2011.
Abstract A discontinuous Galerkin method for the ideal 5 moment two-fluid plasma system is presented. The method uses a second or third order discontinuous Galerkin spatial discretization and a third order TVD Runge-Kutta time stepping scheme. The method is benchmarked against an analytic solution of a dispersive electron acoustic square pulse as well as the two-fluid electromagnetic shock and existing numerical solutions to the GEM challenge magnetic reconnection problem. The algorithm can be generalized to arbitrary geometries and three dimensions. An approach to maintaining small gauge errors based on error propagation is suggested.
10. A H Hakim, J R Cary, J Candy, J Cobb, R H Cohen, T Epperly, D J Estep, S Krasheninnikov, A D Malony, D C McCune, L McInnes, A Pankin, S Balay, J A Carlsson, M R Fahey, R J Groebner, S E Kruger, M Miah, A Pletzer, S Shasharina, S Vadlamani, D Wade-Stein, T D Rognlien, A Morris, S Shende, G W Hammett, K Indireshkumar, A Yu Pigarov, H Zhang. “Coupled whole device simulations of plasma transport in tokamaks with the FACETS code”, SciDAC 2010: J. Physics: Conf. Series, 2010.
Abstract The FACETS project aims to provide computational tools for whole device simulation of tokamak transport for use in fusion applications. The framework provides flexibility by allowing users to choose the best model for a given physics target. Our goals are to develop accurate transport solvers using neoclassical and turbulent fluxes with varying degree of fidelity and computational complexity, including embedded gyrokinetic models. Accurate sources using both ICRH wave absorption and neutral beam injection, using parallel source components, are included. Modeling of the plasma edge using a fluid based component, UEDGE, is performed and coupled to the core solver. The core region is simulated using a newly developed parallel, nested iteration based nonlinear solver while the UEDGE uses nonlinear solves from the PETSc/SNES solver package. As a first application we present coupled core-edge simulations of pedestal buildup in the DIIID tokamak.
9. John Loverich, Ammar Hakim. “Two-Dimensional Modeling of Ideal Merging Plasma Jets”, Journal of Fusion Energy, 29 (6), Pg. 532-539, 2010. Journal link
Abstract Idealized merging argon plasma jets are simulated in 2D using both gas dynamic and MHD models. Results indicate that peak pressures of several hundred kilobar can be achieved for high Mach number jets. Including a simple optically thin Brehmstrahlung radiation model and plasma targets shows that extremely high densities and magnetic fields can be achieved during jet merging on the order of ~1000 times the initial density/field. Further investigations should include detailed ionization processes and more accurate radiation modeling to properly capture the radiation transport and subsequent target compression.
8. John R. Cary, Ammar Hakim, Mahmood Miah, Scott Kruger, Alexander Pletzer, Svetlana Shasharina, Srinath Vadlamani, Alexei Pankin, Ronald Cohen, Tom Epperly, Tom Rognlien, Richard Groebner, Satish Balay, Lois McInnes, Hong Zhang, “FACETS - a Framework for Parallel Coupling of Fusion Components”, The 18th Euromicro International Conference on Parallel, Distributed and Network-Based Computing. Pisa, Italy. 2010.
Abstract Coupling separately developed codes offers an attractive method for increasing the accuracy and fidelity of the computational models. Examples include the earth sciences and fusion integrated modeling. This paper describes the Framework Application for Core-Edge Transport Simulations (FACETS).
Note
This is one of the three papers that describe the proto-Fusion Simulation Programs (proto-FSP). The Facets project aims to tightly couple (with implicit methods) the core-edge and wall regions of the tokamak to perform integrated simulations. The other two proto-FSPs CSWIM and CPES also study coupled systems but they use file/in-memory loose coupling.
7. B. Srinivasan, U. Shumlak and A. Hakim, “Comparisons and Applications of Two-Fluid Plasma Algorithms”, American Institute of Aeronautics and Astronautics. Paper number AIAA 2008-3787, 2008.
This paper describes a study of the five-moment two-fluid plasma model and the asymp- totically approximated fluid model that is derived from it. The two models are compared for applications of an electromagnetic plasma shock, magnetic reconnection and an axisym- metric Z-pinch. The physics captured is compared between these fluid models to determine the regime of applicability. These models are explored for their ability to capture small- scale physics with two-fluid effects that are not sufficiently captured by ideal-MHD.
6. Peter H. Stoltz, Brian Granger, Ammar Hakim, Scott W. Sides and Seth A. Veitzer, “Effects of Sputtering of and Radiation by Aluminum on Magnetized Target Fusion Plasmas”, Journal of Fusion Energy, 27 (1-2), 2008. Pages 119-122.
We estimate numerically the rate of radiation by aluminum impurities for parameters relevant to magnetized target fusion (MTF) plasmas. We demonstrate that the coronal equilibrium is appropriate for expected MTF plasma parameters. Using the coronal equilibrium, we estimate the power radiated per impurity ion is 0.25-0.5:math:10^{16} MW for temperatures and densities relevant to present plasma parameters taken from the FRX-L experiment at Los Alamos National Laboratory and is approximately 75.0:math:10^{16} MW for temperatures and densities relevant to anticipated MTF plasmas. We calculate the sputtering rate of aluminum by thermal deuterium and tritium plasma ions is a few percent assuming an impact angle of 45 degrees. Finally, we estimate that with aluminum impurity levels of a few percent, the impurity radiation power density would be approximately 25 kW/cm3 for FRX-L conditions and 2.5 GW/cm3 for anticipated conditions in a MTF plasma. While we have assumed a sputtering model of impurity generation, the results for the power density apply for impurity levels of a few percent, regardless of the generation mechanism.
5. Ammar H. Hakim, “Extended MHD Modelling with the Ten-Moment Equations”, Journal of Fusion Energy, 27 (1-2), 2008. Pages 36-43.
Abstract High-order moment fluid equations for simulation of plasmas are presented. The ten-moment equations are a two-fluid model in which time dependent equations are used to advance the pressure tensor. With the inclusion of the full pressure tensor Finite Larmor Radius (FLR) effects are captured. Further, Hall-effects are captured correctly by including the full electron momentum equation. Hall and FLR effects are important to understand stability of compact toroids like Field Reversed Configurations (FRCs) and also to detailed understanding of small scale instabilities in current carrying plasmas. The effects of collisions are discussed. Solutions to a Riemann problem for the ten-moment equations are presented. The ten-moment equations show complex dispersive solutions which come about from the source terms. The model is validated with the GEM fast magnetic reconnection challenge problem.
4. A. Hakim, U. Shumlak, “Two-fluid physics and field-reversed configurations”, Physics of Plasmas, 14, 055911, 2007.
Abstract In this paper, algorithms for the solution of two-fluid plasma equations are presented and applied to the study of field-reversed configurations (FRCs). The two-fluid model is more general than the often used magnetohydrodynamic (MHD) model. The model takes into account electron inertia, charge separation, and the full electromagnetic field equations, and it allows for separate electron and ion motion. The algorithm presented is the high-resolution wave propagation scheme. The wave propagation method is based on solutions to the Riemann problem at cell interfaces. Operator splitting is used to incorporate the Lorentz and electromagnetic source terms. The algorithms are benchmarked against the Geospace Environmental Modeling Reconnection Challenge problem. Equilibrium of FRC is studied. It is shown that starting from a MHD equilibrium produces a relaxed two-fluid equilibrium with strong flows at the FRC edges due to diamagnetic drift. The azimuthal electron flow causes lower-hybrid drift instabilities (LHDI), which can be captured if the ion gyroradius is well resolved. The LHDI is known to be a possible source of anomalous resistivity in many plasma configurations. LHDI simulations are performed in slab geometries and are compared to recent experimental results.
3. A. Hakim, J. Loverich, U. Shumlak, “A high resolution wave propagation scheme for ideal Two-Fluid plasma equations”, Journal of Computational Physics, 219, 2006.
Abstract Algorithms for the solution of the five-moment ideal Two-Fluid equations are presented. The ideal Two-Fluid model is more general than the often used magnetohydrodynamic (MHD) model. The model takes into account electron inertia effects, charge separation and the full electromagnetic field equations and allows for separate electron and ion motion. The algorithm presented is the high resolution wave propagation method. The wave propagation method is based on solutions to the Riemann problem at cell interfaces. Operator splitting is used to incorporate the Lorentz and electromagnetic source terms. To preserve the divergence constraints on the electric and magnetic fields two different approaches are used. In the first approach Maxwell equations are rewritten in their mixed-potential form. In the second approach the so-called perfectly hyperbolic form of Maxwell equations are used which explicitly incorporate the divergence equations into the time stepping scheme. The algorithm is applied to a one-dimensional Riemann problem, ion-acoustic soliton propagation and magnetic reconnection. In each case Two-Fluid physics described by the ideal Two-Fluid model is highlighted.
2. Ammar H. Hakim, Brian D. Piening, and Norman J. McCormick, “Near-asymptotic angle dependence of ocean optical radiance”, Applied Optics, 43 (31), 2004.
Abstract The approach of ocean optical radiance to an approximate asymptotic dependence with increasing depth in spatially uniform waters is numerically examined for a variety of sea surface illumination conditions.
1. Ammar H. Hakim and Norman J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters”, Applied Optics, 42 (6), 2003.
Abstract We propose and test an inverse ocean optics procedure with numerically simulated data for the determination of inherent optical properties using in-water radiance measurements. If data are available at only one depth within a deep homogeneous water layer, then the single-scattering albedo and the single parameter that characterizes the Henyey-Greenstein phase function can be estimated. If data are available at two depths, then these two parameters can be determined along with the optical thickness so that the absorption and scattering coefficients, and also the backscattering coefficient, can be estimated. With a knowledge of these parameters, the albedo and Lambertian fraction of reflected radiance of the bottom can be determined if measurements are made close to the bottom. A simplified method for determining the optical properties of the water also is developed for only three irradiance-type measurements if the radiance is approximately in the asymptotic regime.