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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Computational Algorithms for Unit Commitment with AC Power Flows (Final Report)

Security-constrained unit commitment (SCUC) is a key component in power system operations. When AC power flow constraints are considered in the SCUC model (AC-SCUC), the problem becomes extremely difficult due to its discrete and non-convex nature, as described in “Grid Optimization Competition Challenge 3 Problem Formulation (GOCC)”. There are four main challenges: (i) Discrete decisions regarding unit online/offline status and start-up/shut-down procedures for every single unit. The number of discrete decision variables increases considerably when a system integrates multiple generators; (ii) Configuration-based combined-cycle formulations, and multi-commodity models that include ramping products, spin/non-spin products, and regulation up/down products. The combined-cycle units introduce additional discrete decision variables and auxiliary service products further complicate the model by connecting multi-commodity products’ continuous and discrete variables; (iii) SCUC models with AC power flow constraints are far more complex due to massive bilinear terms in the large-scale nonlinear power balance equations. The nonlinear power balance equations are further complicated by the discrete step control variables of shunts; (iv) N − 1 contingency analysis. The size of the model increases linearly with the number of contingencies considered, greatly increasing the size of the optimization model. Accordingly, there is an emergent need to develop a robust algorithm capable of deriving a high-quality solution in a short time and passing through contingency tests simultaneously. In this project, we explore innovative techniques to address this challenging problem by integrating advanced polyhedral theory, approximation methods, relaxation strategies, decomposition techniques, and parallel computing. Each technique approaches the problem from a different perspective, leveraging its specific strengths to tackle distinct challenges. Each individual method has demonstrated its effectiveness in the PI’s previous research. Their integration is expected to significantly reduce the computational time required to solve the proposed complex problem. Successful completion of this project has the potential to transform the industry by enhancing optimization solvers capable of handling large-scale day-ahead energy market clearing models within strict time constraints, while incorporating AC power flow constraints. This advancement will lead to reduced overall generation costs and, consequently, increased social welfare.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2018

Asynchronously parallel optimization solver for finding multiple minima

This paper proposes and analyzes an asynchronously parallel optimization algorithm for finding multiple, high-quality minima of nonlinear optimization problems. Our multistart algorithm considers all previously evaluated points when determining where to start or continue a local optimization run. Theoretical results show that, under certain assumptions, the algorithm almost surely starts a finite number of local optimization runs and identifies, or has a single local optimization run converging to, every minimum. The algorithm is applicable to general optimization settings, but our numerical results focus on the case when derivatives are unavailable. In numerical tests, a PYTHON implementation of the algorithm is shown to yield good approximations of many minima (including a global minimum), and this ability scales well with additional resources. Our implementation’s time to solution is shown also to scale well even when the time to evaluate the function evaluation is highly variable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Developing A Fully Implicit Compressible Flow Valve Model in RELAP-7 and Its Application in Short Term BWR Station Black-Out Analyses

This paper presents a set of dynamical models for SRV simulation and their implementation in the RELAP-7 code, which is a new reactor system analysis code being developed at INL. RELAP-7 is based on the MOOSE (the Multi-Physics Object-Oriented Simulation Environment) framework. The compressible flow SRV model covers three different sets of models: fully closed case, subsonic case, and choking case. The RELAP-7 is a fully implicit code and the preconditioned Jacobian-free Newton-Krylov (JFNK) method is used to solve the discretized nonlinear problem. The JFNK method generally needs a good preconditioning scheme to be effective and efficient. There are different ways to precondition the JFNK method. In RELAP-7, two main options are used to obtain the preconditioning matrix: (1) FDP (Finite Difference Preconditioning) option which uses the default finite differencing method provided in the PETSc package. PETSc is the numerical solver software developed by Argonne National Laboratory and used by MOOSE as the numerical solution engine; (2) SMP (Single Matrix Preconditioning) option which requires code developers to provide approximate analytical Jacobian inputs for the important diagonal and off-diagonal terms. Due to the strong discontinuity in the residual vector space when the compressible flow valve model switches from one set of model to another, FDP option cannot guarantee convergence. Instead, accurate analytical Jacobain terms are provided for the compressible flow valve model. With the SMP option selected, the compressible flow valve model can be solved efficiently under different cases. The developed compressible flow valve model has been used for SRV component simulation in several simplified BWR SBO (Station Black-Out) simulations with RELAP-7. This paper only presents cases for short-term SBO in which all the safety systems fail to start and function properly when SBO occurs. Consequently, only the SRVs automatically open and close periodically to control the primary system pressure and discharge the high temperature and high pressure steam into the suppression pool. Although simplified, this case is quite similar to what happened in Fukushima Daiichi Unit 1. Unit 1 had no RCIC system, while the isolation condenser system was believed to be non-functioning, or only available for a very short period of time during the accident. Therefore, the Unit 1 reactor core reached a fuel failure temperature only a few hours into the accident. Two short-term SBO cases will be compared: one with single SRV discharge line which represents all the SRV lines and another case with multi-group SRVs which represent each group of SRV line with different setting points.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2021

Closures and Simulation for Thermal Radiation Transport in Stochastic Media with Nonlinear Temperature Dependence

Because of the practical challenge of rendering very complex realistic spatial structures for numerical work, it is common practice to resort to characterizing such media as stochastic mixtures of materials, ideally parametrized with low order statistics such as the mean, variance, and correlation functions of the now random material properties. This enables realizations of the medium to be repeatedly generated and radiation transport computations to, in principle, be performed for a large ensemble of these realizations to obtain a statistically well-characterized radiation field. Statistical post-processing yields desired quantities such as conditional and unconditional mean radiation flux and probability distributions of transmitted radiation. However, such computations prove expensive for all but the simplest stochastic geometries and are most suited for benchmarking approximate models. The most common approximations lead to homogenized media so that transport computations are required only on a single medium realization but by construct provide only limited statistical information on the radiation field. Almost all approximate approaches to this problem attempt to develop equations for low order moments of the radiation intensity (mean, second moment, correlation function) but inevitably encounter a closure problem: the equation for any statistical moment will contain terms depending on unknown higher-order moments. Thus, the challenge shifts to one of developing closure relations that relate the unknown moments to the lower order moments. Under very special conditions, an exact closure can be derived but in general closures are heuristically stated constitutive relations. Also, closure approaches depend on whether the mixing statistics are spatially and/or temporally continuous as in fluctuating turbulent fields, or discontinuous as in randomly mixed solid chunks of material. Thus, unconditional averaging is generally applied in the former case but conditional averaging is more appropriate when the mixing is discontinuous. In this work, the emphasis is on binary statistical mixtures of immiscible fluids as as such the quantities of interest are averages (flux, temperature) conditioned on the material type.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Analog Systems for Edge Optimization

Over the past decade, analog computing has the subject of substantial research interest providing a path toward improved computational efficiency in the post-Dennard era. Analog matrix vector multiplication (MVM) accelerators provide a popular approach given the ubiquity of MVM operations in numerous applications. However, historically analog computing systems can struggle with applications requiring high precision due to the inherent susceptibility of these systems to analog non-idealities. Therefore, prior work on analog systems has focused either on applications known to be tolerant of limited precision (e.g., neural network inference), or using expensive techniques to emulate high-precision using many analog MVM operations. In this work, we propose an alternative approach. Motivated by recent advances in inexact nonlinear solvers and optimizers, we explore the potential of co-designing optimization algorithms which can take full advantage of the fundamentally inexact analog MVM operations. To enable these co-designed algorithms we also develop a general mathematical theory of the precision and energy efficiency of analog operations, and a new system architecture for tightly-coupled analog and digital computation. Finally, we examine the applicability of analog computing to a wider class of symmetric positive definite systems and find potential in using analog operations as a sparse approximate inverse preconditioner. With these core innovations, this project provides a path toward effectively implementing optimization algorithms on power-constrained autonomous and semi-autonomous systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Dynamic time-warping correction for shifts in ultrahigh resolving power ion mobility spectrometry and structures for lossless ion manipulations

Detection of arrival time shifts between ion mobility spectrometry (IMS) separations can limit achievable resolving power (Rp), particularly when multiple separations are summed or averaged, as commonly practiced. Such variations are more apparent in higher Rp measurements, and are particularly evident in long path length traveling wave structures for lossless ion manipulations (SLIM) IMS due to their typically much longer separation times. Here we explore the utility of a data processing approaches employing linear alignment (LA) and nonlinear dynamic time warping (DTW) of IMS separations to correct for variations between separations, such as due to pressure fluctuations. For multipass SLIM IMS separations, where narrow mobility range measurements have arrival times that can extend to several seconds, the LA approach effectively corrected for such variations, and significantly improvement Rp for summed separations. However, LA was much less effective for high Rp broad mobility range separations, such as obtained with multilevel SLIM IMS. Changes in IMS arrival times ions were observed to be correlated with small pressure changes, with approximately 0.6% relative arrival time shifts being common, sufficient to result in a loss of Rp for summed separations. Comparison of the approaches showed DTW alignment performed similarly to LA when used over a narrow mobility range, but was significantly better (providing narrower peaks and higher signal intensities) for wide mobility range data. We found the DTW approach increased Rp by as much as 115% for measurements in which 50 IMS separations over 2 seconds were summed, and leading to a large improvement in effective Rp. We conclude that DTW is superior to LA for ultrahigh resolution broad mobility range SLIM IMS separations, correcting for ion arrival time shifts regardless of the cause. Our tool is publicly available for use with universal ion mobility format (.UIMF) and text (.txt) files.

Data alignment, dynamic time warping, ion mobility↗
DOE OSTI2019

Cross-verification of neoclassical transport solutions from XGCa against NEO

Radial neoclassical transport from the global gyrokinetic code XGC has been cross-verified against the local drift-kinetic neoclassical code NEO in NEO's local regime using a Grad-Shafranov equilibrium with a circular boundary. XGC shows excellent agreement with NEO, except for a difference of approximately 10% in the ion heat flux. Close investigation reveals that this difference is due to the simplification in NEO to use the same Coulomb logarithm ln Λ to represent all particle species, while XGC uses different formulas for collisions between different species. When XGC uses NEO's Coulomb logarithm, excellent agreement in the ion heat flux is recovered. Here, the present work not only crossverifies the equation solvers in two codes, but also verifies that the gyrokinetic and the drift-kinetic equations agree and that the linearized and nonlinear Coulomb collision operators agree in the mild gradient, local regime.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

An Efficient Time-Domain Model to Simulate Parametric Resonances in a Floating Body Free to Move in Six Degrees of Freedom: Preprint

We present a computationally efficient time-domain model capable of simulating parametric resonances in a floating body in waves. The model assumes all wave forces to be linear, but the inertia and restoring forces acting on the body are expanded to second order in body motions. The simulation speed on a standard computer is approximately 40 times faster than real time. The model is applied to a soft-moored floating axisymmetric body which absorbs energy through heave, but is otherwise free to move in six degrees of freedom. Under certain conditions, we show that the body responds parametrically with large amplitudes not only in surge and pitch, but also in sway, roll, and yaw, provided it is given some small initial displacement in one of these out-of-plane modes. The predictions are confirmed by simulations using state-of-the-art nonlinear Froude-Krylov and computational fluid dynamics models.

parametric resonance↗
DOE OSTI2016

Transition to turbulence and noise radiation in heated coaxial jet flows

Laminar-turbulent transition and noise radiation of a parametrized set of subsonic coaxial jet flows with a hot primary (core) stream are investigated numerically by Large-Eddy Simulation (LES) and direct noise computation. This study extends our previous research on local linear stability of heated coaxial jet flows by analyzing the nonlinear evolution of initially laminar flows disturbed by a superposition of small-amplitude unstable eigenmodes. First, a baseline configuration is studied to shed light on the flow dynamics of coaxial jet flows. Subsequently, LESs are performed for a range of Mach and Reynolds numbers to systematically analyze the influences of the temperature and the velocity ratios between the primary and the secondary (bypass) stream. The results provide a basis for a detailed analysis of fundamental flow-acoustic phenomena in the considered heated coaxial jet flows. Increasing the primary-jet temperature leads to an increase of fluctuation levels and to an amplification of far-field noise, especially at low frequencies. Strong mixing between the cold bypass stream and the hot primary stream as well as the intermittent character of the flow field at the end of the potential core lead to a pronounced noise radiation at an aft angle of approximately 35{sup ∘}. The velocity ratio strongly affects the shear-layer development and therefore also the noise generation mechanisms. Increasing the secondary-stream velocity amplifies the dominance of outer shear-layer perturbations while the disturbance growth rates in the inner shear layer decrease. Already for r{sub mic} > 40R{sub 1}, where r{sub mic} is the distance from the end of the potential core and R{sub 1} is the core-jet radius, a perfect 1/r{sub mic} decay of the sound pressure amplitudes is observed. The potential-core length increases for higher secondary-stream velocities which leads to a shift of the center of the dominant acoustic radiation in the downstream direction.

42 ENGINEERING↗
DOE OSTI2022

Preconditioned least‐squares Petrov–Galerkin reduced order models

Abstract In this article, we introduce a methodology for improving the accuracy and efficiency of reduced order models (ROMs) constructed using the least‐squares Petrov–Galerkin (LSPG) projection method through the introduction of preconditioning. Unlike prior related work, which focuses on preconditioning the linear systems arising within the ROM numerical solution procedure to improve linear solver performance, our approach leverages a preconditioning matrix directly within the minimization problem underlying the LSPG formulation. Applying preconditioning in this way has the potential to improve ROM accuracy for several reasons. First, preconditioning the LSPG formulation changes the norm defining the residual minimization, which can improve the residual‐based stability constant bounding the ROM solution's error. The incorporation of a preconditioner into the LSPG formulation can have the additional effect of scaling the components of the residual being minimized to make them roughly of the same magnitude, which can be beneficial when applying the LSPG method to problems with disparate scales (e.g., dimensional equations, multi‐physics problems). Importantly, we demonstrate that an “ideal preconditioned” LSPG ROM (a ROM in which the preconditioner is the inverse of the Jacobian of its corresponding full order model) emulates projection of the full order model solution increment onto the reduced basis. This quantity defines a lower bound on the error of a ROM solution for a given reduced basis. By designing preconditioners that approximate the Jacobian inverse—as is common in designing preconditioners for solving linear systems—it is possible to obtain a ROM whose error approaches this lower bound. The proposed approach is evaluated on several mechanical and thermo‐mechanical problems implemented within the Albany HPC code and run in the predictive regime, with prediction across material parameter space. We demonstrate numerically that the introduction of simple Jacobi, Gauss‐Seidel, and ILU preconditioners into the proper orthogonal decomposition/LSPG formulation reduces significantly the ROM solution error, the reduced Jacobian condition number, the number of nonlinear iterations required to reach convergence, and the wall time (thereby improving efficiency). Moreover, our numerical results reveal that the introduction of preconditioning can deliver a robust and accurate solution for test cases in which the unpreconditioned LSPG method fails to converge.

Lindsay, Payton↗
DOE OSTI2025

Recent Widespread Deceleration of Global Surface Urban Heat Islands Unveiled by Satellites

Tracking the temporal dynamics of urban heat island (UHI) is critical for urban heat adaptation and mitigation strategies. However, whether UHI trends have shifted recently and their underlying drivers remain unknown. Here we investigate the variabilities in surface UHI trends and their associated determinants in 2,104 cities worldwide from 2000 to 2022. Our findings reveal that approximately half of the world's cities have experienced notable shifts in surface UHI trends, predominantly characterized by UHI deceleration. These shifts can be primarily attributed to alterations in vegetation trends during the day and to modifications in surface albedo and local warming trends at night. Our study challenges the conventional linear models commonly employed to estimate surface UHI trends, suggesting potential biases in such estimates. Our findings underscore the need for nuanced policies to curtail UHI growth by considering changes in urban underlying surfaces and background climate, particularly from a nonlinear perspective.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2016

PetIGA: A framework for high-performance isogeometric analysis

We present PetIGA, a code framework to approximate the solution of partial differential equations using isogeometric analysis. PetIGA can be used to assemble matrices and vectors which come from a Galerkin weak form, discretized with Non-Uniform Rational B-spline basis functions. We base our framework on PETSc, a high-performance library for the scalable solution of partial differential equations, which simplifies the development of large-scale scientific codes, provides a rich environment for prototyping, and separates parallelism from algorithm choice. We describe the implementation of PetIGA, and exemplify its use by solving a model nonlinear problem. To illustrate the robustness and flexibility of PetIGA, we solve some challenging nonlinear partial differential equations that include problems in both solid and fluid mechanics. Lastly, we show strong scaling results on up to 4096 cores, which confirm the suitability of PetIGA for large scale simulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Structure Tuning, Strong Second Harmonic Generation Response, and High Optical Stability of the Polar Semiconductors Na 1- x K x AsQ 2

We report the mixed cation compounds Na 1-x K x AsSe 2 (x = 0.8, 0.65, 0.5) and Na 0.1 K 0.9 AsS 2 crystallize in the polar noncentrosymmetric space group Cc. The AAsQ(2) (A = alkali metals, Q = S, Se) family features one-dimensional (1D) 1 / ∞ [AQ 2 - ] chains comprising corner-sharing pyramidal AQ 3 units in which the packing of these chains is dependent on the alkali metals. The parallel 1 / ∞ [AQ(2) - ] chains interact via short As ∙∙∙Se contacts, which increase in length when the fraction of K atoms is increased. The increase in the As ∙∙∙Se interchain distance increases the band gap from 1.75 eV in γ-NaAsSe 2 to 2.01 eV in Na 0.35 K 0.65 AsSe 2 , 2.07 eV in Na 0.2 K 0.8 AsSe 2 , and 2.18 eV in Na 0.1 K 0.9 AsS 2 . The Na 1-x K x AsSe 2 (x = 0.8, 0.65) compounds melt congruently at approximately 316 °C. Wavelength-dependent second harmonic generation (SHG) measurements on powder samples of Na 1-x K x AsSe 2 (x = 0.8, 0.65, 0.5) and Na 0.1 K 0.9 AsS 2 suggest that Na 0.2 K 0.8 AsSe 2 and Na 0.1 K 0.9 AsS 2 have the highest SHG response and exhibit significantly higher laser-induced damage thresholds (LIDTs). Theoretical SHG calculations on Na 0.5 K 0.5 AsSe 2 confirm its SHG response with the highest value of d 33 = 22.5 pm/V χ 333 (2) = 45.0 pm/V). The effective nonlinearity for a randomly oriented powder is calculated to be d eff = 18.9 pm/V χ eff (2) = 37.8 pm/V), which is consistent with the experimentally obtained value of d eff = 16.5 pm/V χ eff (2) = 33.0 pm/V). Three-photon absorption is the dominant mechanism for the optical breakdown of the compounds under intense excitation at 1580 nm, with Na 0.2 K 0.8 AsSe 2 exhibiting the highest stability.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Improving the nonlinear control performance of the supply fan at air handling units using a gain scheduling control strategy

Due to its nonlinear nature, the supply fan at air handling units with the controller tuned at the design condition tends to be aggressive and oscillate under partial load conditions. Here, the objective of this paper is to develop and validate a gain scheduling control strategy to improve its nonlinear control performance. First, a control-oriented model, which does not require numerous physical parameters and extensive test data, is developed to study the nonlinearity of the fan system. Based on the theoretical model and experimental verifications, the issue of an aggressive response with a conventional fixed-gain controller is caused by the fact that the system gain is proportional to the ratio of the duct static pressure to the fan speed. To address the issue, a scheduling function of the measurable duct static pressure and fan speed is proposed to be included in the conventional fixed-gain controller to compensate for the fan system gain variation. The gain scheduling control strategy is found to approximately maintain the identical control performance under all operation conditions. Most importantly, the gain scheduling control strategy can be readily implemented without intensive computation and additional measurements, showing a promising potential in industrial applications.

42 ENGINEERING↗
DOE OSTI2015

Shock compression modeling of metallic single crystals: comparison of finite difference, steady wave, and analytical solutions

Background: The shock response of metallic single crystals can be captured using a micro-mechanical description of the thermoelastic-viscoplastic material response; however, using a such a description within the context of traditional numerical methods may introduce a physical artifacts. Advantages and disadvantages of complex material descriptions, in particular the viscoplastic response, must be framed within approximations introduced by numerical methods. Methods: Three methods of modeling the shock response of metallic single crystals are summarized: finite difference simulations, steady wave simulations, and algebraic solutions of the Rankine-Hugoniot jump conditions. For the former two numerical techniques, a dislocation density based framework describes the rate- and temperature-dependent shear strength on each slip system. For the latter analytical technique, a simple (two-parameter) rate- and temperature-independent linear hardening description is necessarily invoked to enable simultaneous solution of the governing equations. For all models, the same nonlinear thermoelastic energy potential incorporating elastic constants of up to order 3 is applied. Results: Solutions are compared for plate impact of highly symmetric orientations (all three methods) and low symmetry orientations (numerical methods only) of aluminum single crystals shocked to 5 GPa (weak shock regime) and 25 GPa (overdriven regime). Conclusions: For weak shocks, results of the two numerical methods are very similar, regardless of crystallographic orientation. For strong shocks, artificial viscosity affects the finite difference solution, and effects of transverse waves for the lower symmetry orientations not captured by the steady wave method become important. The analytical solution, which can only be applied to highly symmetric orientations, provides reasonable accuracy with regards to prediction of most variables in the final shocked state but, by construction, does not provide insight into the shock structure afforded by the numerical methods.

36 MATERIALS SCIENCE↗
DOE OSTI2022

The nonlinear rheology of complex yield stress foods

Many foods have a yield stress that allows them to retain a desired shape at rest, but transition into a viscous fluid when being served or consumed. The determination of the yield stress of the food dictates how the foods are formed and packaged, how they are served, and how they are perceived when being eaten. Oscillatory shearing provides an ideal test protocol to map the rheology across a range of time and flow strength scales. We couple oscillatory shearing and an iterative recovery procedure to show that the yielding process is a continuous transition for two common yield stress foods. We show that unrecoverable processes from oscillatory tests are equivalent to the steady shear flow behavior. We show that this yielding behavior can be well approximated by a recently published model that treats yield stress materials as continuous viscoelastic fluids with a rate-dependent relaxation time and has parameters that can be obtained from the linear viscoelastic oscillatory frequency sweep and the steady shear flow curve.

Griebler, James J.↗
DOE OSTI2022

Dynamic Modeling, Parameter Estimation, and Data Reconciliation of a Supercritical Pulverized Coal-Fired Boiler

A high-fidelity, spatially distributed dynamic model of a supercritical pulverized coal-fired boiler was developed for calculating transient thermal profiles of the boiler tubes, which are often not measured or unmeasurable in industrial settings due to the harsh operating conditions. In this work, rigorous models of the subcritical and supercritical steam properties, which can be highly nonlinear, were used to capture the transition between two-phase and one-phase flow, respectively, as supercritical boilers frequently transition through the critical point during load-following operation. The transient boiler model was also used for data reconciliation and parameter estimation for the purpose of model validation against industrial data. An approach was developed to reduce the size of large-scale reconciliation problem by focusing on the implementation of bias terms for the reconciled variables, which can be estimated in place of the reconciled variables themselves. In addition, a functional approximation of the reconciliation problem was defined so that it can be more simply optimized. Combining this functional approximation optimization with reformulation of the reconciliation problem considering measurement biases can significantly improve the tractability of many such large-scale problems. The validated boiler model was then used to study transient responses under load-following operation with specific attention on the calculation of unmeasurable variables.

42 ENGINEERING↗
DOE OSTI2019

Linear response of a Hall magnetic drift wave for verification of Hall MHD algorithms

Numerical implementations of Hall magnetohydrodynamics (Hall MHD) can be challenging due to the nonlinear multidimensional nature of the Hall term. Here, a model problem is presented that couples the hydrodynamic motion of the plasma to Hall MHD evolution of the magnetic field. The Hall MHD equations are linearized about unperturbed solutions in both cylindrical and Cartesian coordinates in two dimensions. The magnetic field is assumed to lie in the ignorable direction, and the linear response about the unperturbed solution is considered. The resulting ordinary differential equation is used to numerically compute the eigenfunctions and eigenfrequencies of the mode. The resulting eigenfunctions do not make the local wave approximation but are instead global solutions that depend on the spatial dependence of the unperturbed Alfvén speed. Hall MHD simulations are then performed in the Ares multiphysics code and shown to agree with the predicted phase velocities of the wave, and the simulated solutions are shown to numerically converge to the semianalytic modes. By varying the background density of the plasma (and correspondingly, the ion inertial length), the importance of Hall physics can be varied. This allows the test problem to transition from the classical MHD limit to the extreme Hall MHD limit. Furthermore, this problem is a useful tool for the verification of Hall MHD routines implemented in various codes, and the robustness of a routine can be tested in regimes in which Hall physics is dominant.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗