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DOE OSTI2025

Insights from Femtosecond Transient Absorption Spectroscopy into the Structure–Function Relationship of Glyceline Deep Eutectic Solvents

This study aimed to determine the structure–function relationship (SFR) for ChCl–glycerol mixtures, a deep eutectic solvent (DES), by investigating their microscopic solvation dynamics and how it relates to their macroscopic properties across varying concentrations of ChCl. Femtosecond transient absorption (fs-TA) spectroscopy revealed two distinct solvation dynamics time constants: τ 1 , governed by glycerol–glycerol interactions, and τ 2 , dominated by the choline response. The τ 2 minimum at 25–30 mol % ChCl closely aligned with the eutectic composition (~33.33 mol % ChCl), where the glycerol network was the most organized and the choline ions exhibited the fastest relaxation. The viscosity decreased sharply up to ~25 mol % ChCl and then plateaued, while the conductivity increased monotonically with ChCl concentration, reflecting enhanced ionic mobility. The density decreased with both increasing ChCl concentration and temperature, indicating disrupted hydrogen bonding and reduced molecular packing. The polarity, measured using betaine-30 (B30) and the E T (30) polarity scale, increased steeply up to approximately 25 mol % ChCl before reaching a plateau. These findings identified the eutectic composition as the optimal concentration range for balancing stability, fluidity, conductivity, and enhanced dynamics within the glycerol system.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1985

Study of second order upwind differencing in a recirculating flow

The accuracy and stability of the second order upwind differencing scheme was investigated. The solution algorithm employed is based on a coupled solution of the nonlinear finite difference equations by the multigrid technique. Calculations have been made of the driven cavity flow for several Reynolds numbers and finite difference grids. In comparison with the hybrid differencing, the second order upwind differencing is somewhat more accurate but it is not monotonically accurate with mesh refinement. Also, the convergence of the solution algorithm deteriorates with the use of the second order upwind differencing.

Vanka, S. P.↗

NASA NTRS1987

Second-order upwind differencing in a recirculating flow

Vanka, S. P.↗

DOE OSTI2024

High-accuracy method for modeling nucleation and growth of particles

State-of-the-art numerical models describing the kinetics of aerosol particle nucleation and growth from a cooling vapor primarily use a nodal method, in which particles that are smaller than the critical size are omitted from consideration because they are thermodynamically unfavorable. This omission is based on the assumption that most newly formed particles are above the critical size, so that subcritical-size particles are not important to take into account. Due to the nature of the nodal method, it suffers from numerical diffusion, which can cause an artificial broadening of the cluster size distribution leading to a significant overestimation of the number of large-size particles. To address these issues, we propose a more accurate numerical method that explicitly models particles of all sizes, and uses a special numerical scheme that substantially reduces the numerical diffusion and provides high solution accuracy and numerical stability. We extensively compare this novel method to the commonly used nodal solver of the general dynamic equation (GDE) for particle growth and demonstrate that it offers GDE solutions with higher accuracy with low numerical diffusion. Incorporating small subcritical clusters into the solution is crucial for: 1) more precise determination of the entire particle size distribution function and 2) wider applicability of the model to experimental studies with non-monotonic temperature variations leading to particle evaporation. The computational code implementing this numerical method in Python is available upon request.

42 ENGINEERING↗

NASA NTRS2015

A Numerical and Experimental Study of Coflow Laminar Diffusion Flames: Effects of Gravity and Inlet Velocity

In this work, the influence of gravity, fuel dilution, and inlet velocity on the structure, stabilization, and sooting behavior of laminar coflow methane-air diffusion flames was investigated both computationally and experimentally. A series of flames measured in the Structure and Liftoff in Combustion Experiment (SLICE) was assessed numerically under microgravity and normal gravity conditions with the fuel stream CH4 mole fraction ranging from 0.4 to 1.0. Computationally, the MC-Smooth vorticity-velocity formulation of the governing equations was employed to describe the reactive gaseous mixture; the soot evolution process was considered as a classical aerosol dynamics problem and was represented by the sectional aerosol equations. Since each flame is axisymmetric, a two-dimensional computational domain was employed, where the grid on the axisymmetric domain was a nonuniform tensor product mesh. The governing equations and boundary conditions were discretized on the mesh by a nine-point finite difference stencil, with the convective terms approximated by a monotonic upwind scheme and all other derivatives approximated by centered differences. The resulting set of fully coupled, strongly nonlinear equations was solved simultaneously using a damped, modified Newton's method and a nested Bi-CGSTAB linear algebra solver. Experimentally, the flame shape, size, lift-off height, and soot temperature were determined by flame emission images recorded by a digital camera, and the soot volume fraction was quantified through an absolute light calibration using a thermocouple. For a broad spectrum of flames in microgravity and normal gravity, the computed and measured flame quantities (e.g., temperature profile, flame shape, lift-off height, and soot volume fraction) were first compared to assess the accuracy of the numerical model. After its validity was established, the influence of gravity, fuel dilution, and inlet velocity on the structure, stabilization, and sooting tendency of laminar coflow methane-air diffusion flames was explored further by examining quantities derived from the computational results.

microgravity↗

NASA NTRS1991

Nonlinear development and secondary instability of Gortler vortices in hypersonic flows

In a hypersonic boundary layer over a wall of variable curvature, the region most susceptible to Goertler vortices is the temperature adjustment layer over which the basic state temperature decreases monotonically to its free stream value. Except for a special wall curvature distribution, the evolution of Goertler vortices trapped in the temperature adjustment layer will in general be strongly affected by the boundary layer growth through the O(M sup 3/2) curvature of the basic state, where M is the free stream Mach number. Only when the local wavenumber becomes as large as of order M sup 3/8, do nonparallel effects become negligible in the determination of stability properties. In the latter case, Goertler vortices will be trapped in a thin layer of O(epsilon sup 1/2) thickness which is embedded in the temperature adjustment layer; here epsilon is the inverse of the local wavenumber. A weakly nonlinear theory is presented in which the initial nonlinear development of Goertler vortices in the neighborhood of the neutral position is studied and two coupled evolution equations are derived. From these, it can be determined whether the vortices are decaying or growing depending on the sign of a constant which is related to wall curvature and the basic state temperature.

Fu, Yibin B.↗

NASA NTRS1989

The secondary flow and its stability for natural convection in a tall vertical enclosure

The multicellular flow between two vertical parallel plates is numerically simulated using a time-splitting pseudospectral method. The steady flow of air and the time-periodic flow of oil are investigated, and descriptions of these flows using both physical and spectral approaches are presented. The time dependence of the flow and temperature fields of oil are shown, and the dynamics of the process is discussed. The spectral transfer of energy among the axial modes comprising the flow is explored. The three-dimensional linear stabiltiy of the multicellular air flow is parametrically studied. The domain of stable two-dimensional cellular motion is found to be constrained by the Eckhaus instability and by two types of monotone instability. The two-dimensional multicellular flow is unstable above a Grashof number of about 8550.

Chait, Arnon↗

NASA NTRS1997

Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains

Borrowing from techniques developed for conservation law equations, numerical schemes which discretize the Hamilton-Jacobi (H-J), level set, and Eikonal equations on triangulated domains are presented. The first scheme is a provably monotone discretization for certain forms of the H-J equations. Unfortunately, the basic scheme lacks proper Lipschitz continuity of the numerical Hamiltonian. By employing a virtual edge flipping technique, Lipschitz continuity of the numerical flux is restored on acute triangulations. Next, schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous Hamiltonians. These schemes possess a discrete maximum principle on arbitrary triangulations and naturally exhibit proper Lipschitz continuity of the numerical Hamiltonian. Finally, a class of Petrov-Galerkin approximations are considered. These schemes are stabilized via a least-squares bilinear form. The Petrov-Galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy.

Barth, Timothy J.↗

DOE OSTI2025

Modifications in the charge trap landscape in Hf 0.5 Zr 0.5 O 2 as a function of oxygen vacancy concentration observed with photoemission electron microscopy

Oxygen vacancies in Hf x Zr (1-x) O 2 (HZO) both contribute to stabilization of the ferroelectric orthorhombic phase and promote leakage pathways that limit the endurance of devices based on the material. Here, for this reason, the defect states of oxygen vacancies were investigated using photoemission electron microscopy (PEEM) and photoluminescence spectroscopy (PL), as their concentration was varied via ex situ laser exposure. Following a controlled oxygen vacancy reduction via visible (2.54 eV) laser dosing of HZO, deep-ultraviolet (DUV, 5.82 eV) PEEM was used to spatially probe the resulting mid-gap defect states and work function. Work function was found to increase monotonically with the laser-induced reduction in oxygen vacancy concentration culminating in a total increase near 70 meV. The change implies a Fermi level shift toward the valence band as the total available electron-filled charge states are reduced with the removal of oxygen vacancies. A reduction in charge states is corroborated by the observed lessening of both photoemission and photoluminescence intensities after laser dosing. The deduced position of the Fermi level is within a band of near-conduction band defect states produced by oxygen vacancies that are linked to endurance limiting leakage currents. Together, these results directly identify the primary role of oxygen vacancies on the defect states in HZO while demonstrating that laser exposure can be used for their modification.

42 ENGINEERING↗

NASA NTRS1994

The heat capacity of titanium disilicide

The heat capacity of TiSi2 has been measured in the temperature range 105-500 K. The heat capacity of TiSi2 varies monotonically between temperatures of 100 and 500 K with a reference value of 22.0 +/- 0.2 J/g atom K at 298.15 K. Based upon our heat capacity data, the standard molar entropy of TiSi2 at 298.15 K estimated to be 22.2 +/- 0.8 J/g atom K. Our data support estimates of the higher temperature heat capacity of TiSi2 based upon previously measured heat capacities of different, but similar, substances. A number of TiSi2 samples were prepared by rapidly quenching (greater than 10(exp 5) K/s) from the melt. The structure and the measured heat capacity of these samples were similar to those of well annealed samples, underscoring the thermal stability of this material.

Sylla, W. K.↗

DOE OSTI2025

Anomalous softening of 3D printed elastomeric foam irradiated under compressive strain

Elastomeric foam is an essential component in many industrial and technological settings, primarily as thermal insulators and as positional/mechanical support cushions. In particular, silicone foam is utilized in harsh environments due to exceptional thermal and chemical stability. Under service conditions within certain applications such material gets exposed to a high dosage of gamma radiation, which can permanently alter the material’s structural and mechanical response properties. Most studies on gamma-exposure under inert or oxidative atmosphere indicate hardening of silicone foam, which is attributed to radiation-induced enhancement in chemical cross-linking. Here we report two contrasting effects depending on whether (non-oxidative) radiation exposure is carried out with the foam under zero or finite compressive strain. While in the former case we observe radiation-hardening consistent with previous studies, in the latter case (50% porous foam under 30% uniaxial compression) we see a monotonic decrease in Young’s modulus with increasing dosage, although solvent swelling experiments on the constituent rubber indicate a net increase in cross-link density independent of the state of strain. We quantitatively model all dose-dependent data using the Ogden Hyperfoam strain-energy function within the framework of Tobolsky two-network scheme and attribute the above anomaly to a combined effect of radiation-induced thickness change (compression set) and inherent nonlinearity in the foam’s stress-strain response.

Coarse-grained models↗

DOE OSTI2025

Stability and convergence of nuclear detonations in white dwarf collisions

We investigate the numerical stability of thermonuclear detonations in 1D accelerated reactive shocks and 2D binary collisions of equal-mass magnetized and unmagnetized white dwarf stars. To achieve high resolution at initiation sites, we devised geometric gridding and mesh velocity strategies specially adapted to the unique requirements of head-on collisional geometries, scenarios in which one expects maximum production of iron-group products. We study the effects of grid resolution and the limiting of temperature, energy, and reactants for different stellar masses, separations, magnetic fields, inigenerationtial compositions, detonation mechanisms, and limiter parameters across a range of cell sizes from 1 to 100 km. Our results set bounds on the parameter space of limiter amplitudes for which both temperature- and energy-limiting procedures yield consistent and monotonically convergent solutions. Within these bounds, we find that grid resolutions of 5 km or better are necessary for uncertainties in total released energy and iron-group products to drop below 10%. Intermediate-mass products (e.g., calcium) exhibit similar convergence trends but with somewhat greater uncertainty. These conclusions apply equally to pure C/O white dwarfs, multispecies compositions (including helium shells), magnetized and unmagnetized cores, and either single or multiple detonation scenarios.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2026

Methanol Steam Reforming on Ru/m-ZrO 2 : Sodium Promotion of the CO 2 -Forming Pathway

Sodium (Na) promotion of Ru/m-ZrO 2 was investigated to elucidate how an alkali modification tunes selectivity in methanol steam reforming (MSR). H 2 -TPR/XANES/EXAFS show that Na increases surface basicity and strengthens Ru–O interactions, shifting RuO x reduction and H 2 spillover to a higher temperature. DRIFTS reveals Na-induced red shifts of the formate ν(CH) band and changes in OCO vibrational splitting, consistent with weakening of the formate C–H bond and an altered binding geometry. CO 2 -TPD confirms a monotonic shift toward stronger basic sites with increasing Na concentrations. Under MSR conditions, Na selectively increases CO2 concentration at the expense of CO. At ~80% conversion and 325 °C, CO 2 selectivity increases from 12.0% (unpromoted) to 16.2, 21.0, and 26.5% for 0.5, 1.0, and 1.8% Na, respectively; at ~300 °C and ~66–69% conversion, CO 2 selectivity increases from 8.6% to 23.7% at 1.8% Na. Transient MSR experiments further show earlier and larger H 2 evolution upon Na addition, corroborating the promotion of the dehydrogenation/decarbonylation route to CO 2 + H 2 . We propose that Na increases basicity and modifies the Ru–support interface to favor formate dehydrogenation/decarboxylation, thereby increasing the H 2 yield and lowering CO formation. Ru’s higher-energy, less occupied d-band stabilizes CO and oxygenated intermediates more strongly in the reforming environment, making the CO-forming pathway more resistant to suppression than on Pt.

CO2 selectivity↗

NASA NTRS2002

Progress Toward Generation of a Navier-Stokes Database for a Harrier in Ground Effect

The Harrier YAV-8B aircraft is capable of vertical and short-field take-off and landing (V/STOL) by directing its four exhaust nozzles toward the ground, or conventional flight by rotating its nozzles into a horizontal position. The British Royal Air Force and the United States Marine Corps have used this aircraft for more than 30 years to provide a quick reaction time for troop support, and reduce the need for long runways. The success of this powered-lift (PL) vehicle has also prompted the more recent design of the Joint Strike Fighter (JSF). However there are significant safety issues that must be addressed when operating a PL vehicle in close proximity to the ground. Hot Gas Ingestion (HGI) by the inlets can result in a rapid loss of powered lift; and high-speed jet flows along the ground plane can induce low pressures underneath the vehicle, causing a 'suck-down' effect. Under these conditions, departure from controlled flight may occur. Moreover, unsteady ground vortices and jet fountains can affect the aircraft,s controllability and its proximity to ground troops. The viscous, time-dependent flow fields of PL vehicles are difficult to accurately and efficiently predict using Computational Fluid Dynamics (CFD). A number of researchers have used the time-dependent Reynolds-averaged Navier-Stokes (RANS) equations to compute flows for single and multiple jets in a cross-flow. A few have added some geometric complexity to the problem by computing flows for jet-augmented delta wings near a ground plane. Smith et.al. computed for the first time a single RANS solution about a simplified Harrier. This geometry included a fuselage, wing, leading edge root extension (LERX), inlets, and exhaust nozzles. All of these investigations cite two practical problems with computing these flows: 1) the need for improved solution accuracy; and, 2) the need for faster solution methods. We view the need for faster solution methods as key to improving the solution accuracy and making this class of computation more routine. One can hardly refine grids, explore the use of advanced turbulence models, and generate databases when it takes weeks of dedicated computer time for a single solution. Chaderjian, Ahmad, Pandya, and Murman have focused on reducing the time-to-solution for this very difficult and complex problem through process automation and exploitation of parallel computing. They began with the Harrier geometry reported, and added a deflected wing flap and empennage for greater realism. To date more than 80 solutions have been carried out. This paper will describe this process and progress made in reducing the time required to generate a simple longitudinal force and moment database for a Harrier in ground effect. It shows a typical snap-shot from an unsteady streakline animation, where fluid particles are colored by temperature. The ground vortex and a jet-fountain vortex are highlighted. It also shows a similar streakline image, where HGI occurs due to the vehicle in close proximity to the ground. It is show the mean lift coefficient as a function of angle of attack and height. The angle of attack range was 4 deg less than or = alpha less than or = 10 deg with an increment of 1 degree, and the height range was 10 ft less than or = h less than or = 30ft with an increment of 5 feet. This 35 solution database was extended to over 2500 cases using a monotone cubic-spline interpolation procedure. The suck-down effect (reduction of lift near the ground) is highlighted in the figure. The "cushion effect," the conventional reduction of lift as the vehicle moves out of ground effect, is also indicated. All 35 RANS solutions were obtained using 952 Silicon Graphics Origin 2000 and 3000 processors in dedicated mode for one week. Typically, 112 processors were assigned to each case. Some other cases used fewer processors to utilize all available CPUS. The final paper will report on the automation of the solution process, including: grid generation, job monitoring, solution completion criteria, and post processing. Moreover, improvements in parallel efficiency for a dual time-step algorithm for the RANS equations will also be presented. Results will be discussed in detail using unsteady streakline flow visualization to correlate unsteady flow structures with dominant aerodynamic frequencies. The stability derivatives, CL, and CL, will also be presented.

Chaderjian, Neal M.↗

DOE OSTI2026

Input specific neural networks

Neural networks have emerged as powerful tools for mapping between inputs and outputs. However, their black-box nature limits the ability to encode or impose specific structural relationships between inputs and outputs. Many scientific and engineering problems, such as constitutive modeling in solid mechanics, require networks that can enforce convexity, monotonicity, or other structural constraints to ensure physical consistency. Here, we introduce the Input Specific Neural Network (ISNN), a new architecture that enables multiple, distinct constraints to be imposed on different input subsets for scalar-valued outputs. This framework unifies convex, monotone–convex, monotone, and arbitrary mappings within a single network for the first time. Two ISNN architectures with analytical first- and second-order derivatives are developed. We demonstrate the performance on synthetic toy problems, inverse problems in isotropic hyperelasticity, and finite element simulations. ISNNs achieve improved extrapolation behavior, require fewer invariant inputs than standard input convex networks for polyconvex potentials, and enable significant computational savings via manual differentiation. We also show how ISNNs can be used to learn structural relationships between inputs and outputs via a binary gating mechanism. Particularly, ISNNs are employed to model a homogenized anisotropic free energy potential in a decoupled multiscale setting. The network learns whether or not the potential should be modeled as polyconvex and retains only the relevant layers while using the minimum number of inputs. ISNNs provide a flexible foundation for embedding structural priors into neural networks, enhancing both interpretability and stability. They are broadly applicable across computational mechanics and other scientific domains requiring constrained functional relationships.

Jadoon, Asghar A. [Univ. of Texas, Austin, TX (Uni↗

NASA NTRS2020

Differences in Tropical High Clouds Among Reanalyses: Origins and Radiative Impacts

We examine differences among reanalysis highcloud products in the tropics, assess the impacts of these differences on radiation budgets at the top of the atmosphere and within the tropical upper troposphere and lower stratosphere (UTLS), and discuss their possible origins in the context of the reanalysis models. We focus on the ERA5 (fifthgeneration European Centre for Medium-range Weather Forecasts – ECMWF – reanalysis), ERA-Interim (ECMWF Interim Reanalysis), JRA-55 (Japanese 55-year Reanalysis), MERRA-2 (Modern-Era Retrospective Analysis for Research and Applications, Version 2), and CFSR/CFSv2 (Climate Forecast System Reanalysis/Climate Forecast System Version 2) reanalyses. As a general rule, JRA-55 produces the smallest tropical high-cloud fractions and cloud water contents among the reanalyses, while MERRA-2 produces the largest. Accordingly, long-wave cloud radiative effects are relatively weak in JRA-55 and relatively strong in MERRA-2. Only MERRA-2 and ERA5 among the reanalyses produce tropical-mean values of outgoing long-wave radiation (OLR) close to those observed, but ERA5 tends to underestimate cloud effects, while MERRA-2 tends to overestimate variability. ERA5 also produces distributions of longwave, short-wave, and total cloud radiative effects at the top of the atmosphere that are very consistent with those observed. The other reanalyses all exhibit substantial biases in at least one of these metrics, although compensation between the long-wave and short-wave effects helps to constrain biases in the total cloud radiative effect for most reanalyses. The vertical distribution of cloud water content emerges as a key difference between ERA-Interim and other reanalyses. Whereas ERA-Interim shows a monotonic decrease of cloud water content with increasing height, the other reanalyses all produce distinct anvil layers. The latter is in better agreement with observations and yields very different profiles of radiative heating in the UTLS. For example, whereas the altitude of the level of zero net radiative heating tends to be lower in convective regions than in the rest of the tropics in ERAInterim, the opposite is true for the other four reanalyses. Differences in cloud water content also help to explain systematic differences in radiative heating in the tropical lower stratosphere among the reanalyses. We discuss several ways in which aspects of the cloud and convection schemes impact the tropical environment. Discrepancies in the vertical profiles of temperature and specific humidity in convective regions are particularly noteworthy, as these variables are directly constrained by data assimilation, are widely used, and feed back to convective behaviour through their relationships with thermodynamic stability.

High Clouds↗

DOE OSTI2025

Minimal implicit-solvent coarse-grained simulation of Pluronic block copolymers with ionic liquids

Pluronic block copolymers, composed of poly(ethylene oxide) (PEO) and poly(propylene oxide) (PPO) in a triblock structure (PEO–PPO–PEO), are well known for their amphiphilic character and ability to self‐assemble into micelles in aqueous solution. The addition of ionic liquids (ILs) can further modulate the core–shell structures of these copolymers, influencing their stability, critical micellization temperature, and size. However, fully atomistic simulations often become prohibitively expensive due to the size and complexity of these systems. In this work, coarse‐grained simulations using a minimal implicit‐solvent model were performed to examine how two classes of ILs, namely, 1‐alkyl‐3‐methylimidazolium ([C n C 1 im]) and 1‐alkyl‐3‐methylpyrrolidinium ([C n C 1 pyrr]), change the micellization of Pluronic block copolymers in aqueous solution. The effects of IL concentration and alkyl group length were investigated, and the model greatly improved the efficiency of simulating large‐scale micelle systems. Furthermore, the numerical simulations are qualitatively compared with experimental investigations. Our results show that adding ILs expands the micelle core by embedding IL tails among the PPO blocks, thereby increasing overall micelle size. Less polar ILs generally induce more pronounced micellar growth. However, the effect of IL tail length on conformation and micellar packing is non‐monotonic. Up to moderate chain lengths (around C8–C10), the IL tails can extend sufficiently to increase local separation within the micelle; at longer tail lengths, enhanced hydrophobic clustering and steric hindrance cause the tails to bend or fold, capping further expansion. In addition, although block copolymer chains tend to pack more closely in the presence of longer‐tailed ILs, the random coil size of an individual polymer chain does not necessarily shrink. Meanwhile, these insights provide a deeper understanding of how Pluronic/IL systems interact, informing applications in drug delivery, cosmetics, food, and environmental engineering. Finally, our minimal implicit‐solvent model can be applied to larger systems and longer timescales, substantially reducing computational cost while reproducing key structural trends observed experimentally.

Atomistic simulations↗

NASA NTRS2017

On High-Order Upwind Methods for Advection

In the fourth installment of the celebrated series of five papers entitled "Towards the ultimate conservative difference scheme", Van Leer (1977) introduced five schemes for advection, the first three are piecewise linear, and the last two, piecewise parabolic. Among the five, scheme I, which is the least accurate, extends with relative ease to systems of equations in multiple dimensions. As a result, it became the most popular and is widely known as the MUSCL scheme (monotone upstream-centered schemes for conservation laws). Schemes III and V have the same accuracy, are the most accurate, and are closely related to current high-order methods. Scheme III uses a piecewise linear approximation that is discontinuous across cells, and can be considered as a precursor of the discontinuous Galerkin methods. Scheme V employs a piecewise quadratic approximation that is, as opposed to the case of scheme III, continuous across cells. This method is the basis for the on-going "active flux scheme" developed by Roe and collaborators. Here, schemes III and V are shown to be equivalent in the sense that they yield identical (reconstructed) solutions, provided the initial condition for scheme III is defined from that of scheme V in a manner dependent on the CFL number. This equivalence is counter intuitive since it is generally believed that piecewise linear and piecewise parabolic methods cannot produce the same solutions due to their different degrees of approximation. The finding also shows a key connection between the approaches of discontinuous and continuous polynomial approximations. In addition to the discussed equivalence, a framework using both projection and interpolation that extends schemes III and V into a single family of high-order schemes is introduced. For these high-order extensions, it is demonstrated via Fourier analysis that schemes with the same number of degrees of freedom 𝐾 per cell, in spite of the different piecewise polynomial degrees, share the same sets of eigenvalues and thus, have the same stability and accuracy. Moreover, these schemes are accurate to order 2𝐾−1, which is higher than the expected order of 𝐾.

high-order methods↗