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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 865 records · Page 48
All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

Abort performance for a winged-body single-stage to orbit vehicle

Optimal control theory is employed to determine the performance of abort to orbit (ATO) and return to launch site (RTLS) maneuvers for a single-stage to orbit vehicle. The vehicle configuration examined is a seven engine, winged-body vehicle, that lifts-off vertically and lands horizontally. The abort maneuvers occur as the vehicle ascends to orbit and are initiated when the vehicle suffers an engine failure. The optimal control problems are numerically solved in discretized form via a nonlinear programming (NLP) algorithm. A description highlighting the attributes of this NLP method is provided. ATO maneuver results show that the vehicle is capable of ascending to orbit with a single engine failure at lift-off. Two engine out ATO maneuvers are not possible from the launch pad, but are possible after launch when the thrust to weight ratio becomes sufficiently large. Results show that single engine out RTLS maneuvers can be made for up to 180 seconds after lift-off and that there are scenarios for which RTLS maneuvers should be performed instead of ATP maneuvers.

Lyon, Jeffery A.↗
NASA NTRS2000

Flight Test of a Technology Transparent Light Concentration Panel on SMEX/WIRE

A flight experiment has demonstrated a modular solar concentrator that can be used as a direct substitute replacement for planar photovoltaic panels in spacecraft solar arrays. The Light Concentrating Panel (LCP) uses an orthogrid arrangement of composite mirror strips to form an array of rectangular mirror troughs that reflect light onto standard, high-efficiency solar cells at a concentration ratio of approximately 3:1. The panel area, mass, thickness, and pointing tolerance has been shown to be similar to a planar array using the same cells. Concentration reduces the panel's cell area by 2/3, which significantly reduces the cost of the panel. An opportunity for a flight experiment module arose on NASA's Small Explorer / Wide-Field Infrared Explorer (SMEX/WIRE) spacecraft, which uses modular solar panel modules integrated into a solar panel frame structure. The design and analysis that supported implementation of the LCP as a flight experiment module is described. Easy integration into the existing SMEX-LITE wing demonstrated the benefits of technology transparency. Flight data shows the stability of the LCP module after nearly one year in Low Earth Orbit.

Stern, Theodore G.↗
NASA NTRS1958

Theoretical Calculations of Supersonic Wave Drag at Zero Lift for a Particular Store Arrangement

An analysis, based on the linearized thin-airfoil theory for supersonic speeds, of the wave drag at zero lift has been carried out for a simple two-body arrangement consisting of two wedgelike surfaces, each with a rhombic lateral cross section and emanating from a common apex. Such an arrangement could be used as two stores, either embedded within or mounted below a wing, or as auxiliary bodies wherein the upper halves could be used as stores and the lower halves for bomb or missile purposes. The complete range of supersonic Mach numbers has been considered and it was found that by orienting the axes of the bodies relative to each other a given volume may be redistributed in a manner which enables the wave drag to be reduced within the lower supersonic speed range (where the leading edge is substantially subsonic). At the higher Mach numbers, the wave drag is always increased. If, in addition to a constant volume, a given maximum thickness-chord ratio is imposed, then canting the two surfaces results in higher wave drag at all Mach numbers. For purposes of comparison, analogous drag calculations for the case of two parallel winglike bodies with the same cross-sectional shapes as the canted configuration have been included. Consideration is also given to the favorable (dragwise) interference pressures acting on the blunt bases of both arrangements.

Margolis, Kenneth↗
NASA NTRS2001

Ultraviolet Broad Absorption Features and the Spectral Energy Distribution of the QSO PG 1351+64

We present a moderate-resolution (approximately 20 km s(exp -1) spectrum of the mini broad absorption line QSO PG 1351+64 between 915-1180 A, obtained with the Far Ultraviolet Spectroscopic Explorer (FUSE). Additional low-resolution spectra at longer wavelengths were also obtained with the Hubble Space Telescope (HST) and ground-based telescopes. Broad absorption is present on the blue wings of C III (lambda)977, Ly(beta), O VI (lambda)(lambda)1032,1038, Ly(alpha), N V (lambda)(lambda)1238,1242, Si IV (lambda)(lambda)1393,1402, and C IV (lambda)(lambda)1548,1450. The absorption profile can be fitted with five components at velocities of approximately -780, -1049, -1629, -1833, and -3054 km s(exp -1) with respect to the emission-line redshift of z = 0.088. All the absorption components cover a large fraction of the continuum source as well as the broad-line region. The O VI emission feature is very weak, and the O VI/Ly(alpha) flux ratio is 0.08, one of the lowest among low-redshift active galaxies and QSOs. The UV (ultraviolet) continuum shows a significant change in slope near 1050 A in the restframe. The steeper continuum shortward of the Lyman limit extrapolates well to the observed weak X-ray flux level. The absorbers' properties are similar to those of high-redshift broad absorption-line QSOs. The derived total column density of the UV absorbers is on the order of 10(exp 21) cm(exp -2), unlikely to produce significant opacity above 1 keV in the X-ray. Unless there is a separate, high-ionization X-ray absorber, the QSO's weak X-ray flux may be intrinsic. The ionization level of the absorbing components is comparable to that anticipated in the broad-line region, therefore the absorbers may be related to broad-line clouds along the line of sight.

Zheng, W.↗
NASA NTRS1990

Hypersonic vehicle simulation model: Winged-cone configuration

Aerodynamic, propulsion, and mass models for a generic, horizontal-takeoff, single-stage-to-orbit (SSTO) configuration are presented which are suitable for use in point mass as well as batch and real-time six degree-of-freedom simulations. The simulations can be used to investigate ascent performance issues and to allow research, refinement, and evaluation of integrated guidance/flight/propulsion/thermal control systems, design concepts, and methodologies for SSTO missions. Aerodynamic force and moment coefficients are given as functions of angle of attack, Mach number, and control surface deflections. The model data were estimated by using a subsonic/supersonic panel code and a hypersonic local surface inclination code. Thrust coefficient and engine specific impulse were estimated using a two-dimensional forebody, inlet, nozzle code and a one-dimensional combustor code and are given as functions of Mach number, dynamic pressure, and fuel equivalence ratio. Rigid-body mass moments of inertia and center of gravity location are functions of vehicle weight which is in turn a function of fuel flow.

Shaughnessy, John D.↗
NASA NTRS1999

A Reynolds Number Study of Wing Leading-Edge Effects on a Supersonic Transport Model at Mach 0.3

A representative supersonic transport design was tested in the National Transonic Facility (NTF) in its original configuration with small-radius leading-edge flaps and also with modified large-radius inboard leading-edge flaps. Aerodynamic data were obtained over a range of Reynolds numbers at a Mach number of 0.3 and angles of attack up to 16 deg. Increasing the radius of the inboard leading-edge flap delayed nose-up pitching moment to a higher lift coefficient. Deflecting the large-radius leading-edge flap produced an overall decrease in lift coefficient and delayed nose-up pitching moment to even higher angles of attack as compared with the undeflected large- radius leading-edge flap. At angles of attack corresponding to the maximum untrimmed lift-to-drag ratio, lift and drag coefficients decreased while lift-to-drag ratio increased with increasing Reynolds number. At an angle of attack of 13.5 deg., the pitching-moment coefficient was nearly constant with increasing Reynolds number for both the small-radius leading-edge flap and the deflected large-radius leading-edge flap. However, the pitching moment coefficient increased with increasing Reynolds number for the undeflected large-radius leading-edge flap above a chord Reynolds number of about 35 x 10 (exp 6).

Williams, M. Susan↗
DOE OSTI2016

Radial trends in imf-sensitive absorption features in two early-type galaxies: Evidence for abundance-driven gradients

Samples of early-type galaxies show a correlation between stellar velocity dispersion and the stellar initial mass function (IMF) as inferred from gravity-sensitive absorption lines in the galaxies’ central regions. To search for spatial variations in the IMF, we have observed two early-type galaxies with Keck/LRIS and measured radial gradients in the strengths of absorption features from 4000–5500 Å and 8000–10000 Å. We present spatially resolved measurements of the dwarf-sensitive spectral indices Na I (8190 Å) and Wing-Ford FeH (9915 Å), as well as indices for species of H, C{sub 2}, CN, Mg, Ca, TiO, and Fe. Our measurements show a metallicity gradient in both objects, and Mg/Fe consistent with a shallow gradient in α-enhancement, matching widely observed trends for massive early-type galaxies. The Na I index and the CN{sub 1} index at 4160 Å exhibit significantly steeper gradients, with a break at r∼0.1 r{sub eff} (r∼300 pc). Inside this radius, Na I strength increases sharply toward the galaxy center, consistent with a rapid central rise in [Na/Fe]. In contrast, the ratio of the FeH to Fe index strength decreases toward the galaxy center. This behavior cannot be reproduced by a steepening IMF inside of 0.1 r{sub eff} if the IMF is a single power law. While gradients in the mass function above ∼0.4 M{sub ⊙} may occur, exceptional care is required to disentangle these IMF variations from the extreme variations in individual element abundances near the galaxies’ centers.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2001

Ultraviolet Broad Absorption Features and the Spectral Energy Distribution of the QSO PG 1351+641

We present a moderate-resolution (approximately 20 km/s) spectrum of the broad-absorption line QSO PG 1351+64 between 915-1180 angstroms, obtained with the Far Ultraviolet Spectroscopic Explorer (FUSE). Additional low-resolution spectra at longer wavelengths were also obtained with the Hubble Space Telescope (HST) and ground-based telescopes. Broad absorption is present on the blue wings of C III lambda977, Ly-beta, O VI lambda-lambda-1032,1038, Ly-alpha, N V lambda-lambda-1238,1242, Si IV lambda-lambda-1393,1402, and C IV lambda-lambda-1548,1450. The absorption profile can be fitted with five components at velocities of approximately -780, -1049, -1629, -1833, and -3054 km/s with respect to the emission-line redshift of z = 0.088. All the absorption components cover a large fraction of the continuum source as well as the broad-line region. The O VI emission feature is very weak, and the O VI/Ly-alpha flux ratio is 0.08, one of the lowest among low-redshift active galaxies and QSOs. The ultraviolet continuum shows a significant change in slope near 1050 angstroms in the restframe. The steeper continuum shortward of the Lyman limit extrapolates well to the observed weak X-ray flux level. The absorbers' properties are similar to those of high-redshift broad absorption-line QSOs. The derived total column density of the UV absorbers is on the order of 10(exp 21)/s, unlikely to produce significant opacity above 1 keV in the X-ray. Unless there is a separate, high-ionization X-ray absorber, the QSO's weak X-ray flux may be intrinsic. The ionization level of the absorbing components is comparable to that anticipated in the broad-line region, therefore the absorbers may be related to broad-line clouds along the line of sight.

Zheng, W.↗
NASA NTRS1990

The complex nature of the Seyfert galaxy NGC 7592

Long slit spectra of NGC 7592 were taken on Sep. 26 to 30, 1989 at the 1.52 cm European Southern Observatory (ESO) telescope, equipped with a Boller and Chivens spectrograph and an RCA High Resolution charge coupled device (CCD) camera. The problem of the nature of Region C is addressed at first. C shows an heliocentric radial velocity very similar to that of Regions A and B. Moreover, the arm departing from C is most probably a tidal tail, because its extension is large and its orientation is peculiar. The high H alpha luminosity of C is typical of a starburst nucleus. These facts argue in favor of C being the nucleus of a third galactic component (southern component S) physically interacting with the SE component of NGC 7592. The directions of the velocity vectors in various regions of NGC 7592 are marked. It is noteworthy that the SE component rotates clockwise, if the radial velocity difference delta v sub r from its nucleus B is due to rotation. Under the same assumption for the delta v sub r = v sub r-v sub r, A, the NW component seems to rotate counterclockwise. Thus, the gas in the regions where the two galactic bodies are in contact moves in the same way, suggesting that a prograde encounter is occurring. It is known (e.g., Toomre and Toomre, 1972) that prograde encounters have the most disruptive effects on the interacting galaxies, leading to the formation of tidal tails. The interpretation of the wing of the NW component in terms of a tidal tail thus appears very likely. A similar situation holds for the interaction between SE and S too, where S rotates counterclockwise. The interpretation of the arm departing from C as a tidal tail is supported also in this case. The difference in radial velocity between A and B (delta v sub r approx. equal - 40 km s(exp-1)) and the morphology of NGC 7592 suggests that the NW component is beyond the SE one and is approaching it. The most heavily reddened regions (E(B - V) approx. equals 0.7, derived from the H alpha/H beta ratio) are B2 and B4, namely the regions between A and B. Moreover, the small difference in radial velocity between A,B,C suggests that the three galaxies form a bound system, and their kinematics hints that they are most probably in the early stage of a merging phenomenon.

Rafanelli, Piero↗
NASA NTRS1948

Calculations of the Dynamic Stress of Several Airplane Wings in Various Gusts

A series of calculations of the dynamic response of airplane wings to gusts were made with the purpose of showing the relative response of a reference airplane, the DC-3 airplane, and of newer types of airplanes represented by the DC-4, DC-6, and L-49 airplanes. Additional calculations were made for the DC-6 airplane to show the effects of speed and altitude. On the basis of the method of calculation used and the conditions selected for analysis, it is indicated that: 1) The newer airplanes show appreciably greater dynamic stress in gusts then does the reference airplane; 2) Increasing the forward speed or the operating altitude results in an increase of the dynamic stress ratio for the gust with a gradient distance of 10 chords.

Pierce, Harold B.↗
NASA NTRS2015

Plume and Shock Interaction Effects on Sonic Boom in the 1-foot by 1-foot Supersonic Wind Tunnel

The desire to reduce or eliminate the operational restrictions of supersonic aircraft over populated areas has led to extensive research at NASA. Restrictions are due to the disturbance of the sonic boom, caused by the coalescence of shock waves formed by the aircraft. A study has been performed focused on reducing the magnitude of the sonic boom N-wave generated by airplane components with a focus on shock waves caused by the exhaust nozzle plume. Testing was completed in the 1-foot by 1-foot supersonic wind tunnel to study the effects of an exhaust nozzle plume and shock wave interaction. The plume and shock interaction study was developed to collect data for computational fluid dynamics (CFD) validation of a nozzle plume passing through the shock generated from the wing or tail of a supersonic vehicle. The wing or tail was simulated with a wedgeshaped shock generator. This test entry was the first of two phases to collect schlieren images and off-body static pressure profiles. Three wedge configurations were tested consisting of strut-mounted wedges of 2.5- degrees and 5-degrees. Three propulsion configurations were tested simulating the propulsion pod and aft deck from a low boom vehicle concept, which also provided a trailing edge shock and plume interaction. Findings include how the interaction of the jet plume caused a thickening of the shock generated by the wedge (or aft deck) and demonstrate how the shock location moved with increasing nozzle pressure ratio.

Shock wave interaction↗
NASA NTRS1973

Wind tunnel investigation of effects of variations in Reynolds number and leading-edge treatment on the aerodynamic characteristics of an externally blown jet-flap configuration

An investigation has been conducted in a full-scale tunnel to determine the effects of variations in Reynolds number and leading-edge treatment on the aerodynamic characteristics of an externally blown jet-flap transport configuration. The model had a double-slotted trailing-edge flap and was powered by four high-bypass-ratio turbofan engines. Tests were performed by using each of three leading-edge devices (a 30-percent-chord flap and 15- and 25-percent-chord slats) at Reynolds numbers from 0.47 x one million to 1.36 x one million thrust coefficients up to 3.5. The use of a 25-percent-chord slat was found to be more effective than a 15-percent-chord slat or a 30-percent-chord flap in extending the stall angle of attack and in minimizing the loss of lift after the stall. The large slat was also effective in reducing the rolling moments that occurred when the engine-out wing stalled first.

Parlett, L. P.↗
NASA NTRS1976

Measurements in the near-wall region of a relaxing three-dimensional low speed turbulent air boundary layer

An experimental investigation was conducted at selected locations of the near-wall region of a three dimensional turbulent air boundary layer relaxing in a nominally zero external pressure gradient behind a transverse hump (in the form of a 30 deg swept, 5-foot chord wing-type model) faired into the side wall of a low speed wind tunnel. Wall shear stresses measured with a flush-mounted hot-film gage and a sublayer fence were in very good agreement with experimental data obtained with two Preston probes. With the upstream unit Reynolds number held constant at 325,000/ft. approximately one-fourth of the boundary layer thickness adjacent to the wall was surveyed with a single rotated hot-wire probe mounted on a specially designed minimum interference traverse mechanism. The boundary layer (approximately 3.5 in thick near the first survey station where the length Reynolds number was 5.5 million) had a maximum crossflow velocity ratio of 0.145 and a maximum crossflow angle of 21.875 deg close to the wall.

Hebbar, K. S.↗
NASA NTRS1999

Practical Aerodynamic Design Optimization Based on the Navier-Stokes Equations and a Discrete Adjoint Method

Compressible and incompressible versions of a three-dimensional unstructured mesh Reynolds-averaged Navier-Stokes flow solver have been differentiated and resulting derivatives have been verified by comparisons with finite differences and a complex-variable approach. In this implementation, the turbulence model is fully coupled with the flow equations in order to achieve this consistency. The accuracy demonstrated in the current work represents the first time that such an approach has been successfully implemented. The accuracy of a number of simplifying approximations to the linearizations of the residual have been examined. A first-order approximation to the dependent variables in both the adjoint and design equations has been investigated. The effects of a "frozen" eddy viscosity and the ramifications of neglecting some mesh sensitivity terms were also examined. It has been found that none of the approximations yielded derivatives of acceptable accuracy and were often of incorrect sign. However, numerical experiments indicate that an incomplete convergence of the adjoint system often yield sufficiently accurate derivatives, thereby significantly lowering the time required for computing sensitivity information. The convergence rate of the adjoint solver relative to the flow solver has been examined. Inviscid adjoint solutions typically require one to four times the cost of a flow solution, while for turbulent adjoint computations, this ratio can reach as high as eight to ten. Numerical experiments have shown that the adjoint solver can stall before converging the solution to machine accuracy, particularly for viscous cases. A possible remedy for this phenomenon would be to include the complete higher-order linearization in the preconditioning step, or to employ a simple form of mesh sequencing to obtain better approximations to the solution through the use of coarser meshes. An efficient surface parameterization based on a free-form deformation technique has been utilized and the resulting codes have been integrated with an optimization package. Lastly, sample optimizations have been shown for inviscid and turbulent flow over an ONERA M6 wing. Drag reductions have been demonstrated by reducing shock strengths across the span of the wing. In order for large scale optimization to become routine, the benefits of parallel architectures should be exploited. Although the flow solver has been parallelized using compiler directives. The parallel efficiency is under 50 percent. Clearly, parallel versions of the codes will have an immediate impact on the ability to design realistic configurations on fine meshes, and this effort is currently underway.

Grossman, Bernard↗
NASA NTRS1999

Practical Aerodynamic Design Optimization Based on the Navier-Stokes Equations and a Discrete Adjoint Method

The technical details are summarized below: Compressible and incompressible versions of a three-dimensional unstructured mesh Reynolds-averaged Navier-Stokes flow solver have been differentiated and resulting derivatives have been verified by comparisons with finite differences and a complex-variable approach. In this implementation, the turbulence model is fully coupled with the flow equations in order to achieve this consistency. The accuracy demonstrated in the current work represents the first time that such an approach has been successfully implemented. The accuracy of a number of simplifying approximations to the linearizations of the residual have been examined. A first-order approximation to the dependent variables in both the adjoint and design equations has been investigated. The effects of a "frozen" eddy viscosity and the ramifications of neglecting some mesh sensitivity terms were also examined. It has been found that none of the approximations yielded derivatives of acceptable accuracy and were often of incorrect sign. However, numerical experiments indicate that an incomplete convergence of the adjoint system often yield sufficiently accurate derivatives, thereby significantly lowering the time required for computing sensitivity information. The convergence rate of the adjoint solver relative to the flow solver has been examined. Inviscid adjoint solutions typically require one to four times the cost of a flow solution, while for turbulent adjoint computations, this ratio can reach as high as eight to ten. Numerical experiments have shown that the adjoint solver can stall before converging the solution to machine accuracy, particularly for viscous cases. A possible remedy for this phenomenon would be to include the complete higher-order linearization in the preconditioning step, or to employ a simple form of mesh sequencing to obtain better approximations to the solution through the use of coarser meshes. . An efficient surface parameterization based on a free-form deformation technique has been utilized and the resulting codes have been integrated with an optimization package. Lastly, sample optimizations have been shown for inviscid and turbulent flow over an ONERA M6 wing. Drag reductions have been demonstrated by reducing shock strengths across the span of the wing.

Grossman, Bernard↗
DOE OSTI2023

β- and γ-Counting for Pre-detonation Nuclear Forensics on Eu-155

Post-detonation nuclear forensics was performed at Los Alamos National Laboratory (LANL) on 155Eu, a fission product on the wing of the fission product production curve whose yield is sensitive to fission fuel and neutron energy. With a half-life of 4.753 years, 155Eu provides a longer-lived option for these measurements than other fission products with similar mass numbers. The Chemistry Division Group - Nuclear and Radiochemistry, at LANL routinely measures a suite of fission products from 235U fissions in thermal neutron flux experiments known as thermal calibration exercises, using a mixture of gas proportional β-decay counting and γ-spectrometry on HPGe detectors. The fission products of interest are reported relative to a high-yield reference fission product from the same sample to create a running average ratio specific to neutron energy and fuel type; Equation 1 below shows the ratio-of-ratios R-value measured fission products are reported in: R i x = [A( i X)/A( 99 Mo)] unknown [A( i X)/A( 99 Mo)] 235 U n,th where A denotes activity, i X is the nuclide of interest, and 235 U n,th denotes irradiations of 235 U with thermal neutrons. Measurements of unknown fission spectra are ratioed to a running-average of thermal calibration results, and the resulting R-value can be referred to as against a library of irradiation conditions.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS1957

Preliminary Wind-Tunnel Investigation of the Performance of Republic F-105 Wing-Root Inlet Configurations at Various Angles of Attack and a Mach Number of 2.01

A 1/13-scale model of the forebody of the Republic F-105 with twin-duct wing-root inlets was tested in the Langley 4- by 4-foot supersonic pressure tunnel through a range of angle of attack from -4 deg to 15 deg at a Mach number of 2.01 and a Reynolds number of approximately 3.4 x 10(exp 6) per foot. The tests were made with four configurations which incorporated varying amounts of sweep and stagger of the inlet leading edges, modifications to the areas of the boundary-layer diverter floor plate, and modifications to the area of the boundary-layer diverter bleed slots. The highest overall pressure recovery at an angle of attack of 0 deg (average total-pressure recovery, 0.84 mass-flow ratio, 0.98) was achieved with configuration having an inlet leading-edge sweep angle of 58 deg with no leading-edge stagger. Stagger was found to improve the angle-of- attack performance, but at a sacrifice in inlet efficiency for an angle of attack of 0 deg. The boundary-layer diverter floor height, of the order of one boundary-layer thickness, was satisfactory for bypassing the fuselage boundary layer. The boundary-layer diverter-plate bleed slots were effective in increasing the total-pressure recovery of the inlet. The total-pressure-recovery contour plots, taken at the compressor-face station, indicate the existence of high-velocity "cores" throughout the inlet operating range.

Kouyoumjian, W. L.↗
DOE OSTI2020

Scalability of OpenFOAM Density-Based Solver with Runge–Kutta Temporal Discretization Scheme

Compressible density-based solvers are widely used in OpenFOAM, and the parallel scalability of these solvers is crucial for large-scale simulations. In this paper, we report our experiences with the scalability of OpenFOAM’s native rhoCentralFoam solver, and by making a small number of modifications to it, we show the degree to which the scalability of the solver can be improved. The main modification made is to replace the first-order accurate Euler scheme in rhoCentralFoam with a third-order accurate, four-stage Runge-Kutta or RK4 scheme for the time integration. The scaling test we used is the transonic flow over the ONERA M6 wing. This is a common validation test for compressible flows solvers in aerospace and other engineering applications. Numerical experiments show that our modified solver, referred to as rhoCentralRK4Foam, for the same spatial discretization, achieves as much as a 123.2% improvement in scalability over the rhoCentralFoam solver. As expected, the better time resolution of the Runge–Kutta scheme makes it more suitable for unsteady problems such as the Taylor–Green vortex decay where the new solver showed a 50% decrease in the overall time-to-solution compared to rhoCentralFoam to get to the final solution with the same numerical accuracy. Finally, the improved scalability can be traced to the improvement of the computation to communication ratio obtained by substituting the RK4 scheme in place of the Euler scheme. All numerical tests were conducted on a Cray XC40 parallel system, Theta, at Argonne National Laboratory.

Li, Sibo↗