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At least 289 records · Page 16

Isopropanol dehydration reaction rate kinetics measurement using H 2 O time histories

H 2 O formation during thermal decomposition of isopropanol was measured behind reflected shock waves at temperatures ranging from 1127 to 1621 K at an average pressure of 1.42 atm using a laser absorption technique. Of the five modern chemical kinetics models used to compare the H 2 O time histories, the model from Li et al (Combust. Flame 2019;207:171-185) showed the best overall agreement. Sensitivity and rate of production analyses using the Li et al model (as well as those from AramcoMech 3.0, CRECK, and Togbe et al (Energy Fuel. 2011;25:676-683)) showed unimolecular dehydration of isopropanol, iC 3 H 7 OH ⇌ H 2 O + C 3 H 6 (R1), is nearly the sole reaction controlling H2O production at early times, allowing for an a priori measurement of the forward rate constant k 1 . The Arrhenius expression k1 (s –1 ) = 2.60 × 10 13 exp(-31 120 K/T) was determined to represent best the data from this study. According to the models and previous experimental investigations, the pressure investigated is well within the high-pressure limit (HPL) for this reaction. Additional, higher-pressure experiments also confirmed the HPL assumption. An uncertainty analysis was performed by varying secondary reactions within their uncertainties and examining their effect on the overall prediction, establishing an uncertainty within ±20% for all but the highest temperature cases, which have a maximum uncertainty of ±40%. Experiments conducted with a radical trapper, toluene, showed little influence from radical chemistry, suggesting this estimated uncertainty is fairly conservative. Experimental data from Heyne et al (Z Phys Chem. 2015;229:881-907) were found to be in good agreement with the rate measurements from this study and, therefore, a second Arrhenius expression, k 1 (s –1 ) = 2.11 × 10 13 exp(-30 820 K/T), was found to represent both datasets well. This second expression has a larger temperature range of 976-1621 K. Here the present study provides the first high-temperature data collected for this reaction, adding to the limited data available for isopropanol in the literature.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Shock‐tube spectroscopic water measurements and detailed kinetics modeling of 1‐pentene and 3‐methyl‐1‐butene

Abstract To understand the effects of the chemical structure of two C 5 alkene isomers on their combustion properties, and to highlight the major chemical reactions occurring during their high‐temperature oxidation, water time histories were measured behind reflected shock waves for the oxidation of 1‐pentene (C 5 H 10 ‐1) and 3‐methyl‐1‐butene (3M1B) in 99.5% Ar. The experiments were carried out at three different equivalence ratios ( φ = 0.5, 1.0, and 2.0) at pressures and temperatures ranging from 1.29 to 1.47 atm and 1 331 to 1 877 K, respectively. The H 2 O quantification extends the database for 1‐pentene and provides new insights for 3M1B. These unique results were used to validate and to develop a new detailed kinetics model. Numerical predictions are presented, and the new model was able to capture the results with suitable accuracy, with 3M1B being notably more reactive than C 5 H 10 ‐1. Sensitivity and rate‐of‐production analyses were performed to help explain the results. Under the present conditions, the reactivity is rapidly initiated by molecular dissociation of a fraction of the pentene isomers. The initiation phase then induces H‐atom abstraction by active radicals (H, OH, O, HO 2 , and CH 3 ) to first produce alkenyl C 5 H 9 radicals (or an alkyl radical and an alkenyl radical by breaking a C─C bond) and subsequent, smaller fragments. The difference in terms of reactivity between the isomers is essentially due to the fact that 3M1B has one particularly weak tertiary allylic C─H bond, which allows for fast H‐atom abstraction compared with 1‐pentene.

Grégoire, Claire M.↗

Neuroprotectin D1 Attenuates Blast Overpressure Induced Reactive Microglial Cells in the Cochlea

Objective/Hypothesis We examined a neuroinflammatory response associated with glial activation in the cochlea exposed to blast overpressure and evaluated the potential therapeutic efficacy of specialized pro‐resolving mediators such as neuroprotectin D1, NPD1; (10R, 17S‐dihydroxy‐4Z, 7Z, 11E, 13E, 15Z, 19Z‐docosahexaenoic acid) in a rodent blast‐induced auditory injury model. Study Design Animal Research. Methods A compressed‐air driven shock tube was used to expose anesthetized adult male Long‐Evan rats to shock waves simulating an open‐field blast exposure. Approximately 30 minutes after blast exposure, rats were treated with NPD1 (100 ng/kg body wt.) or vehicle delivered intravenously via tail vein injection. Rats were then euthanized 48 hours after blast exposure. Unexposed rats were included as controls. Tissue sections containing both middle and inner ear were prepared with hematoxylin–eosin staining to elucidate histopathological changes associated with blast exposure. Cochlear tissues were evaluated for relative expression of ionized calcium‐binding adaptor 1 (Iba1), as an indicator of microglial activation by immunohistochemistry and western blot analyses. Results Our animal model resulted in an acute injury mechanism manifested by damage to the tympanic membrane, hemorrhage, infiltration of inflammatory cells, and increased expression of Iba1 protein. Moreover, therapeutic intervention with NPD1 significantly reduced Iba1 expression in the cochlea, suggesting a reduction of a neuroinflammatory response caused by blast overpressure. Conclusions Blast overpressure resulted in an increased expression of proteins involved in gliosis within the auditory system, which were reduced by NPD1. Treatment of NPD1 suggests an effective strategy to reduce or halt auditory microglial cell activation due to primary blast exposure. Level of Evidence NA Laryngoscope , 131:E2018–E2025, 2021

Ríos, José David↗

Nozzle-Free Printing of CNT Electronics Using Laser-Generated Focused Ultrasound

Printed electronics have made remarkable progress in recent years and inkjet printing (IJP) has emerged as one of the leading methods for fabricating printed electronic devices. However, challenges such as nozzle clogging, and strict ink formulation constraints have limited their widespread use. To address this issue, a novel nozzle-free printing technology is explored, which is enabled by laser-generated focused ultrasound, as a potential alternative printing modality called Shock-wave Jet Printing (SJP). Specifically, the performance of SJP-printed and IJP-printed bottom-gated carbon nanotube (CNT) thin film transistors (TFTs) is compared. While IJP required ten print passes to achieve fully functional devices with channel dimensions ranging from tens to hundreds of micrometers, SJP achieved comparable performance with just a single pass. For optimized devices, SJP demonstrated six times higher maximum mobility than IJP-printed devices. Furthermore, the advantages of nozzle-free printing are evident, as SJP successfully printed stored and unsonicated inks, delivering moderate electrical performance, whereas IJP suffered from nozzle clogging due to CNT agglomeration. Moreover, SJP can print significantly longer CNTs, spanning the entire range of tube lengths of commercially available CNT ink. The findings from this study contribute to the advancement of nanomaterial printing, ink formulation, and the development of cost-effective printable electronics.

47 OTHER INSTRUMENTATION↗

Gluon double-spin asymmetry in the longitudinally polarized p + p collisions

We derive the first-ever small-x expression for the inclusive gluon production cross section in the central rapidity region of the longitudinally polarized proton-proton collisions. The cross section depends on the polarizations of both protons, therefore comprising the numerator of the longitudinal double-spin asymmetry ALL for the produced gluons. The cross section is calculated in the shock wave formalism and is expressed in terms of the polarized dipole scattering amplitudes on the projectile and target protons. We show that the small-x evolution corrections are included into our cross section expression if one evolves these polarized dipole amplitudes using the double-logarithmic helicity evolution derived in [1–4]. Our calculation is performed for the gluon sector only, with the quark contribution left for future work. When that work is complete, the resulting formula will be applicable to longitudinally polarized proton-proton and proton-nucleus collisions, as well as to polarized semi-inclusive deep inelastic scattering (SIDIS) on a proton or a nucleus. Our results should allow one to extend the small-x helicity phenomenology analysis of [5] to the jet/hadron production data reported for the longitudinally polarized proton-proton collisions at RHIC and to polarized SIDIS measurements at central rapidities to be performed at the EIC.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Quark and gluon helicity evolution at small x: revised and updated

We revisit the problem of small Bjorken-x evolution of the gluon and flavor-singlet quark helicity distributions in the shock wave (s-channel) formalism. Earlier works on the subject in the same framework resulted in an evolution equation for the gluon field-strength F 12 and quark “axial current” $\overline{\psi}\gamma$ + γ 5 ψ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (summing powers of α s ln 2 (1/x) with α s the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, ${}_D{}^{\leftarrow i} {}_D{}^{i}$, also sandwiched between the light-cone Wilson lines (with the repeated transverse index i = 1, 2 summed over), was missing in the previous works. This operator has the physical meaning of the sub-eikonal (covariant) phase: its contribution to helicity evolution is shown to be proportional to another sub-eikonal operator, ${}_D{}^{i} {}_D{}^{\leftarrow i}$, which is related to the Jaffe-Manohar polarized gluon distribution. In this work we include this new operator into small-x helicity evolution, and construct novel evolution equations mixing all three operators (${}_D{}^{i} {}_D{}^{\leftarrow i}$, F 12 , and $\overline{\psi}\gamma$ + γ 5 ψ), generalizing the results of . We also construct closed double-logarithmic evolution equations in the large-N c and large-N c &N f limits, with N c and N f the numbers of quark colors and flavors, respectively. Solving the large-N c equations numerically we obtain the following small-x asymptotics of the quark and gluon helicity distributions ΔΣ and ΔG, along with the g 1 structure function, $$ \Delta \Sigma \left(x,{Q}^2\right)\sim \Delta G\left(x,{Q}^2\right)\sim {g}_1\left(x,{Q}^2\right)\sim {\left(\frac{1}{x}\right)}^{3.66\sqrt{\frac{\alpha_s{N}_c}{2\pi }}} $$ in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Low and moderate x gluon contribution to exclusive Compton scattering processes

We revisit the high energy semi-classical description of the exclusive processes DVCS, TCS, and Double DVCS by explicitly keeping track of the Feynman x dependence in both the hard and the hadronic matrix elements. This is achieved by a modification of the standard shock wave approximation to derive the effective Feynman rules, which leads to a generic expression on which we then perform a partial twist expansion to get rid of quantities suppressed by the proper physical scales. We obtain a compact factorized master formula that can be used to investigate the Bjorken limit at leading twist. In particular, we recover the full one-loop result in the collinear limit for pure gluon exchange with the target. Finally, we discuss the subtleties in taking the simultaneous collinear and small x limit.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Dijet impact factor in DIS at next-to-leading order in the Color Glass Condensate

In this work, we compute the next-to-leading order impact factor for inclusive dijet production in deeply inelastic electron-nucleus scattering at small x Bj . Our computation, performed in the framework of the Color Glass Condensate effective field theory, includes all real and virtual contributions in the gluon shock wave background of all-twist lightlike Wilson line correlators. We demonstrate explicitly that the rapidity evolution of these correlators, to leading logarithmic accuracy, is described by the JIMWLK Hamiltonian. When combined with the next-to-leading order JIMWLK Hamiltonian, our results for the impact factor improve the accuracy of the inclusive dijet cross-section to \( \mathcal{O} \) ( \( {\alpha}_s^2 \) ln( x f /x Bj )), where x f is a rapidity factorization scale. These results are an essential ingredient in assessing the discovery potential of inclusive dijets to uncover the physics of gluon saturation at the Electron-Ion Collider.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Modification to the Hawking temperature of a dynamical black hole by a flow-induced supertranslation

One interesting proposal to solve the black hole information loss paradox without modifying either general relativity or quantum field theory, is the soft hair, a diffeomorphism charge that records the anisotropic radiation in the asymptotic region. This proposal, however, has been challenged, given that away from the source the soft hair behaves as a coordinate transformation that forms an Abelian group, thus unable to store any information. To maintain the spirit of the soft hair but circumvent these obstacles, we consider Hawking radiation as a probe sensitive to the entire history of the black hole evaporation, where the soft hairs on the horizon are induced by the absorption of a null anisotropic flow, generalizing the shock wave considered in [1, 2]. To do so we introduce two different time-dependent extensions of the diffeomorphism associated with the soft hair, where one is the backreaction of the anisotropic null flow, and the other is a coordinate transformation that produces the Unruh effect and a Doppler shift to the Hawking spectrum. Together, they form an exact BMS charge generator on the entire manifold that allows the nonperturbative analysis of the black hole horizon, whose surface gravity, i.e. the Hawking temperature, is found to be modified. The modification depends on an exponential average of the anisotropy of the null flow with a decay rate of 4M, suggesting the emergence of a new 2-D degree of freedom on the horizon, which could be a way out of the information loss paradox.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

The SuperCam Instrument Suite on the NASA Mars 2020 Rover: Body Unit and Combined System Tests

The SuperCam instrument suite provides the Mars 2020 rover, Perseverance, with a number of versatile remote-sensing techniques that can be used at long distance as well as within the robotic-arm workspace. These include laser-induced breakdown spectroscopy (LIBS), remote time-resolved Raman and luminescence spectroscopies, and visible and infrared (VISIR; separately referred to as VIS and IR) reflectance spectroscopy. A remote micro-imager (RMI) provides high-resolution color context imaging, and a microphone can be used as a stand-alone tool for environmental studies or to determine physical properties of rocks and soils from shock waves of laser-produced plasmas. SuperCam is built in three parts: The mast unit (MU), consisting of the laser, telescope, RMI, IR spectrometer, and associated electronics, is described in a companion paper. The on-board calibration targets are described in another companion paper. Here we describe SuperCam’s body unit (BU) and testing of the integrated instrument. The BU, mounted inside the rover body, receives light from the MU via a 5.8 m optical fiber. The light is split into three wavelength bands by a demultiplexer, and is routed via fiber bundles to three optical spectrometers, two of which (UV and violet; 245–340 and 385–465 nm) are crossed Czerny-Turner reflection spectrometers, nearly identical to their counterparts on ChemCam. The third is a high-efficiency transmission spectrometer containing an optical intensifier capable of gating exposures to 100 ns or longer, with variable delay times relative to the laser pulse. This spectrometer covers 535–853 nm ( $105\text{--}7070~\text{cm}^{-1}$ Raman shift relative to the 532 nm green laser beam) with $12~\text{cm}^{-1}$ full-width at half-maximum peak resolution in the Raman fingerprint region. The BU electronics boards interface with the rover and control the instrument, returning data to the rover. Thermal systems maintain a warm temperature during cruise to Mars to avoid contamination on the optics, and cool the detectors during operations on Mars.Results obtained with the integrated instrument demonstrate its capabilities for LIBS, for which a library of 332 standards was developed. Examples of Raman and VISIR spectroscopy are shown, demonstrating clear mineral identification with both techniques. Luminescence spectra demonstrate the utility of having both spectral and temporal dimensions. Finally, RMI and microphone tests on the rover demonstrate the capabilities of these subsystems as well.

47 OTHER INSTRUMENTATION↗

Physical, numerical, and computational challenges of modeling neutrino transport in core-collapse supernovae

The proposal that core collapse supernovae are neutrino driven is still the subject of active investigation more than 50 years after the seminal paper by Colgate and White. The modern version of this paradigm, which we owe to Wilson, proposes that the supernova shock wave is powered by neutrino heating, mediated by the absorption of electron-flavor neutrinos and antineutrinos emanating from the proto-neutron star surface, or neutrinosphere. Neutrino weak interactions with the stellar core fluid, the theory of which is still evolving, are flavor and energy dependent. The associated neutrino mean free paths extend over many orders of magnitude and are never always small relative to the stellar core radius. Thus, neutrinos are never always fluid like. Instead, a kinetic description of them in terms of distribution functions that determine the number density of neutrinos in the six-dimensional phase space of position, direction, and energy, for both neutrinos and antineutrinos of each flavor, or in terms of angular moments of these neutrino distributions that instead provide neutrino number densities in the four-dimensional phase-space subspace of position and energy, is needed. In turn, the computational challenge is twofold: (i) to map the kinetic equations governing the evolution of these distributions or moments onto discrete representations that are stable, accurate, and, perhaps most important, respect physical laws such as conservation of lepton number and energy and the Fermi–Dirac nature of neutrinos and (ii) to develop efficient, supercomputer-architecture-aware solution methods for the resultant nonlinear algebraic equations. In this review, we present the current state of the art in attempts to meet this challenge.

79 ASTRONOMY AND ASTROPHYSICS↗

Adaptive mesh based combustion simulations of direct fuel injection effects in a supersonic cavity flame-holder

Here, we present high-fidelity reacting simulations of a supersonic cavity flame-holder configuration. The focus of this work is on flame stabilization brought about by varying the location of fuel injection in a cavity stabilized supersonic flow of air. Central to our approach is a compressible multi-species reacting flow solver that uses adaptive-mesh-refinement (AMR), enabling the resolution of flame, shock-waves, boundary-layers, and small-scale structures in the computational domain. Our analysis indicates that fuel injection closer to the ramp at the aft end of the cavity allows for greater mixing and lower peak temperatures compared to fuel injection upstream that is closer to the backward facing step of the cavity. This difference is mainly due to greater turbulent fluctuations generated from the shear-layer towards the cavity ramp, thereby enhancing the mixing of fuel and air. A low frequency oscillatory behaviour in heat-release and pressure was also observed for the upstream injection case while a much higher-frequency phenomena was observed in the near-ramp injection case. By identifying the important physical determinants of the combustion processes, this study illustrates a promising pathway to design and optimize direct fuel injection strategies in supersonic cavity flame-holders that can improve flame stability, combustion efficiency, and reduce emissions.

42 ENGINEERING↗

3D mesh regularization within an ALE code using a weighted line sweeping method

The Lagrangian formalism is widely used to simulate hydrodynamic responses in complex engineering applications, particularly those involving strong shock waves. However, as the mesh moves with the fluid, it can become highly distorted, requiring a regularization step. This involves constructing a new grid and remapping conservative quantities onto it to restore mesh quality. This work introduces a regularization method for block-structured meshes within a 3D ALE (Arbitrary Lagrangian-Eulerian) code. The proposed approach prevents mesh tangling while preserving the anisotropic features of the initial Lagrangian mesh. This regularization technique incorporates aspect ratio-based weights to control mesh smoothing. Unlike uniform rezoning techniques, this weighted approach maintains proximity to the Lagrangian mesh while improving mesh quality. Here, the method effectively handles concave geometries by mitigating the grid attraction phenomenon, which typically leads to mesh concentration along concave edges. Numerical experiments demonstrate its efficiency in regularizing severely deformed meshes, and its integration within the ALE framework is validated on challenging hydrodynamic test cases, including the triple point problem.

42 ENGINEERING↗

Enhanced ductility in in-layer glass-carbon fiber/epoxy hybrid composites produced via tailored fiber placement

Experimental mechanical property results of composite materials with hybrid reinforcements (commingled glass and carbon fibers) are detailed and compared with single-fiber composite properties. In-layer hybrid materials, with carbon and glass fiber tows laid side-by-side, were produced via tailored fiber placement (TFP) technology. Detailed experiments showed a phenomenon of “enhanced ductility” of carbon fiber in the hybrid composites of 12–24% relative to an all-carbon fiber composite and multiple stress–strain peaks were observed. Here, this enhanced ductility was hypothesized to be due to the glass fibers mitigating the shock waves arising from the initial failures of carbon fibers, and preventing the premature failure of the remaining carbon fibers. A novel way to engineer the stress–strain behavior of a hybrid composite to achieve a metal-like ductile response (plateau of stress–strain behavior, often termed “elastic–plastic deformation”) was demonstrated by carefully selecting the type and composition of carbon and glass fiber materials.

36 MATERIALS SCIENCE↗

Measurements of sound propagation in Mars' lower atmosphere

Acoustics has become extraterrestrial and Mars provides a new natural laboratory for testing sound propagation models compared to those ones on Earth. Owing to the unique combination of a microphone and two sound sources, the Ingenuity helicopter and the SuperCam laser-induced sparks, the Mars 2020 Perseverance rover payload enables the in situ characterization of unique sound propagation properties of the low-pressure CO 2 -dominated Mars atmosphere. In this study, we show that atmospheric turbulence is responsible for a large variability in the sound amplitudes from laser-induced sparks. This variability follows the diurnal pattern of turbulence. In addition, acoustic measurements acquired over one Martian year reveal a variation of the sound intensity by a factor of 1.8 from a constant source due to the seasonal cycle of pressure and temperature that significantly modifies the acoustic impedance and shock-wave formation. Finally, we show that the evolution of the Ingenuity tones and laser spark amplitudes with distance is consistent with one of the existing sound absorption models, which is a key parameter for numerical simulations applied to geophysical experiments on CO 2 -rich atmospheres. Overall, these results demonstrate the potential of sound propagation to interrogate the Mars environment and will therefore help in the design of future acoustic-based experiments for Mars or other planetary atmospheres such as Venus and Titan.

79 ASTRONOMY AND ASTROPHYSICS↗

Gaseous jet through an outward opening injector: Details of mixing characteristic and turbulence scales

Direct injection (DI) strategy of natural gas (NG) into internal combustion engines (ICE) has led to higher thermal efficiency and lower exhaust emissions. In order to thoroughly understand the most relevant phenomena affecting the performances of such engines, computational fluid dynamics (CFD) plays a key role as an accurate description of the jet evolution and interaction within the combustion chamber is required to that aim. Accurate description of high-pressure gaseous jets is rather challenging at high Mach numbers, as the injected gas is strongly under-expanded once in the ambient, giving room to shocks due to compressibility effects. Also the interaction between shock waves and mixing layers needs to be carefully represented with a multi-dimensional model, calling for substantial computational resources requirements. In this paper a numerical investigation of the behavior of a gaseous jet (Argon) through an outward opening injector has been carried out. A Large Eddy Simulation (LES) approach has been used in order to track the structures derived by the interaction of the injected fuel with the surrounding ambient. Although already good results were obtained using a Reynolds Averaged Navier-Stokes (RANS) approach, the adoption of LES is required to characterize more accurately the jet properties in terms of vortex structures and mixing effectiveness. Finally, the effect of the Nozzle Pressure Ratio (NPR) on the jet evolution has been highlighted in the paper, showing how a higher NPR would give a faster injection process, compromising however the homogeneity of the mixture.

33 ADVANCED PROPULSION SYSTEMS↗

Droplet breakup by multimodal nonlinear Rayleigh Taylor instability

A droplet impacted by a shock wave will undergo a process of fragmentation due to the development of interfacial hydrodynamic instabilities. The interface experiences variable acceleration and shear that result in the development of both inertial (Rayleigh–Taylor) and shear (Kelvin–Helmholtz) instabilities. These perturbations grow in time and drive the fragmentation and breakup of the deformed droplet. Experiments are performed on nominally 0.86 mm water droplet subjected to a Mach 7.6 detonation wave, resulting in a high Weber number (~36,000) breakup event. Perturbation growth is measured from a series of high-speed (> MHz) shadowgraph images. It is proposed that, given the size of the large-scale perturbations observed in experiments, these instabilities are growing in the non-linear regime and can be described by bubble-merger models for nonlinear mixing. Calculations are performed for the growth rates and size of these instabilities using deformation and external flow models to establish the time-dependent boundary conditions. The concurrence of the measured perturbation widths and the predictions of the simple model lend credence to the theory. This novel approach serves to open a new avenue in the characterization of droplet breakup via hydrodynamic instabilities.

Droplet breakup↗

An efficient, conservative, time-implicit solver for the fully kinetic arbitrary-species 1D-2V Vlasov-Ampère system

In this paper, we consider the solution of the fully kinetic (including electrons) Vlasov-Ampère system in a one-dimensional physical space and two-dimensional velocity space (1D-2V) for an arbitrary number of species with a time-implicit Eulerian algorithm. The problem of velocity-space meshing for disparate thermal and bulk velocities is dealt with by an adaptive coordinate transformation of the Vlasov equation for each species, which is then discretized, including the resulting inertial terms. Mass, momentum, and energy are conserved, and Gauss's law is enforced to within the nonlinear convergence tolerance of the iterative solver through a set of nonlinear constraint functions while permitting significant flexibility in choosing discretizations in time, configuration, and velocity space. We mitigate the temporal stiffness introduced by, e.g., the plasma frequency through the use of high-order/low-order (HOLO) acceleration of the iterative implicit solver. We present several numerical results for canonical problems of varying degrees of complexity, including the multiscale ion-acoustic shock wave problem, which demonstrate the efficacy, accuracy, and efficiency of the scheme.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗