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Results for “PARABOLIC EQUATION”

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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13 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

The Parabolic Wave Equation Generalized to a Spatially Varying Plasma in a Vertical Magnetic Field

This report walks through the process of deriving the wave equation for a vertical wave being propagated through a spatially varying plasma in a vertical magnetic field. Additionally, this report shows how to derive the index of refraction of the plasma corresponding to three different modes of the vertical magnetic field. Finally, this paper briefly explores the next steps planned to successfully simulate and test the equations in a pre-existing phase screen scintillation code.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Effect of a Fine-Scale Layered Structure of the Atmosphere on Infrasound Signals from Fragmenting Meteoroids

We investigate the influence of a fine-scale (FS) layered structure in the atmosphere on the propagation of infrasound signals generated by fragmenting meteoroids. Using a pseudo-differential parabolic equation (PPE) approach, we model broadband acoustic signals from point sources at altitudes of 35–100 km. The presence of FS fluctuations in the stratosphere (37–45 km) and the lower thermosphere (100–120 km) modifies ray trajectories, causing multiple arrivals and prolonged signal durations at ground stations. In particular, meteoroids fragmenting at 80–100 km can produce two distinct thermospheric arrivals beyond 150 km range, while meteoroids descending to 50 km or below yield weak, long-lived arrivals within the acoustic shadow zone via antiguiding propagation and diffraction. Comparison with observed infrasound data confirms that FS-layered inhomogeneities can account for multi-arrival “N-waves,” broadening potential interpretations of meteoroid signals. The results also apply to other atmospheric-entry objects, such as sample return capsules, emphasizing how FS structure impacts shock wave propagation. In conclusion, our findings advance understanding of wavefield evolution in a layered atmosphere and have broad relevance for global infrasound monitoring of diverse phenomena (e.g., re-entry capsules, rocket launches, and large-scale explosions).

Aeroacoustics↗
DOE OSTI2026

Multi-arrival infrasound from meteoroids: Fragmentation signatures versus propagation effects in a fine-scale layered atmosphere

Infrasonic signatures of meteoroid fragmentation are frequently ambiguous: do multiple arrivals signify a complex breakup or merely the distorting effects of a layered atmosphere? Resolving this ambiguity is critical for accurate energy estimates and source reconstruction. In this study, we address this challenge by analyzing a unique regional dataset of well-constrained meteoroid events observed by the Southern Ontario Meteor Network and the co-located Elginfield Infrasound Array. We employ pseudo-differential parabolic equation (PPE) simulations to quantify how fine-scale gravity-wave structures in the stratosphere and lower thermosphere modify acoustic waveforms at ranges <300 km. Our modeling reveals that while fine-scale layering can stretch signals and generate diffuse oscillatory tails, it does not produce discrete, high-amplitude pulse splitting at ranges below ∼140 km. By applying these results to the rare multi-arrival event 20060305, we demonstrate that its distinct double arrival at 100 km range is inconsistent with atmospheric multipathing and provides definitive evidence of separate fragmentation episodes. These findings establish new diagnostic criteria for separating source physics from propagation artifacts, improving the reliability of infrasound as a monitoring tool for natural bolides, space debris re-entries, and catastrophic launch failures.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Block Island Acoustic Propagation Modeling Data

This dataset contains acoustic propagation model outputs, computational subroutines, and analysis tools for underwater sound propagation in the Block Island region, including parabolic equation (PE) model results, visualization products, and comprehensive modeling software tools.

17 WIND ENERGY↗
DOE OSTI2026

Boundary-layer receptivity to oblique freestream vorticity waves for a high-enthalpy hypersonic flow

The receptivity of a Mach 15 straight-cone boundary layer to oblique freestream vorticity waves is investigated using direct numerical simulation (DNS) alongside linear stability theory and the linear parabolized stability equations. A thermochemical nonequilibrium gas model is used. Oblique freestream vorticity waves at frequencies of 400, 800, and 1200 kHz are considered, with incident angles ranging from 0° to 29.4° at 400 kHz, 0° to 15.7° at 800 kHz, and 0° to 20.6° at 1200 kHz. The 400 kHz case is of primary interest due to the strong second-mode amplification at this frequency. Although the underlying base flow is axisymmetric, the oblique vorticity waves lead to a fully three-dimensional boundary-layer disturbance whose characteristics vary depending on the azimuthal ray relative to the freestream wave. Moving downstream within the second-mode instability region, some clear trends emerge in terms of the boundary-layer disturbance amplitudes; that is, disturbance amplitudes are highest at the leeward ray (relative to the freestream wave), but weakest about halfway between the windward and leeward rays. Increasing the incident angle causes the amplitudes to increase on the leeward ray and decrease on the windward ray. Moreover, the boundary-layer disturbance throughout contains a wide spectrum of azimuthal wavenumbers in which the disturbance energy falls off at higher wavenumbers. Increasing the incident angle causes the azimuthal spectrum of the boundary-layer disturbance to broaden overall. Qualitatively similar results are found for the two higher frequencies leading up to the peak-amplitude locations corresponding to the second-mode instability.

42 ENGINEERING↗
DOE OSTI2025

Doppler Broadening and Other Temperature Effects

In this paper I attempt to document what I have learned and still remember about temperature effects, particularly Doppler broadening, on nuclear data, as it applies to ENDF formatted data. My focus is on the SIGMA1 method of Doppler broadening that I developed over 50 years ago. However, I do realize that there are currently many different methods used in computer codes, and I have tried to keep my discussion general as it applies to ALL of these methods. In particular, ALL of the MYTHS I describe below apply to all Doppler broadening methods: they all solve the parabolic diffusion equation in spherical geometry in (speed, reaction rate) versus temperature. These methods differ only in how the cross section is represented. So, at 85 years old here is what I still remember.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Reduced basis approximations of parameterized dynamical partial differential equations via neural networks

Projection-based reduced order models are effective at approximating parameter-dependent differential equations that are parametrically separable. When parametric separability is not satisfied, which occurs in both linear and nonlinear problems, projection-based methods fail to adequately reduce the computational complexity. Devising alternative reduced order models is crucial for obtaining efficient and accurate approximations to expensive high-fidelity models. In this work, we develop a timestepping procedure for dynamical parameter-dependent problems, in which a neural-network is trained to propagate the coefficients of a reduced basis expansion. This results in an online stage with a computational cost independent of the size of the underlying problem. Here, we demonstrate our method on several parabolic partial differential equations, including a problem that is not parametrically separable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Multiscale Modeling and Experimental Insights into High-Temperature Soil Biodegradation Dynamics of Semi-Crystalline Poly(Lactic Acid) Nonwoven Fabrics

This study investigates the biodegradation of semi-crystalline poly(lactic acid) (PLA) nonwovens (NWs) in soil at 58 °C using both experimental and mathematical modeling approaches. The model utilizes a system of parabolic diffusion-reaction partial differential equations (PDEs) to elucidate chemical transformations over time and in space. It accounts for phenomena such as the diffusion of water and lactic acid monomers through the polymer matrix and into the surrounding soil, along with their microbial breakdown. It also accounts for the initial PLA crystallinity and predicts its evolution in time. The model is solved numerically for a single filament, and the results were used to shed light on PLA NW transformations observed in soil over a 180-day incubation period. Various characterization techniques, including scanning electron microscopy (SEM), differential scanning calorimetry (DSC), and Raman spectroscopy, were employed to assess morphological changes, crystallinity, and molecular changes in the NWs throughout the experiment. By comparing the experimental data with the model predictions, the hydrolysis rate coefficient was found to be 3.37 × 10 -7 s -1 , while the rate of microbial degradation of lactic acid monomers was faster, of the order of 9.63 × 10 -7 s -1 . The findings highlight the significant role of crystallinity in the biodegradation process. The PLA degradation ceases when no amorphous material remains, and the crystallinity reaches 0.8, as observed in the experiments by day 120. Furthermore, this research contributes to a deeper understanding of PLA biodegradation dynamics and offers insights for effectively managing biodegradable materials in environmental settings.

Diffusion−reaction modeling↗
DOE OSTI2025

Direct Discontinuous Galerkin methods for the reacting multi-component flow equations

The Direct Discontinuous Galerkin (DDG (Liu and Yan, 2008)) method and a counterpart with Interface Correction (DDGIC (Danis and Yan, 2022)) are extended to compute diffusion terms that arise when solving the compressible multi-component flow equations in thermochemical nonequilibrium. Thermodynamic properties, transport properties, chemical reaction rates, and energy exchange terms are computed using Mutation++ (Scoggins et al., 2020). The DG method is applied on unstructured grids, where the accuracy and convergence rates can be sensitive to the numerical method chosen for parabolic terms. A method for determining the homogeneity tensor of the flow equations required for DDGIC is shown. The convergence properties of the DDG methods are studied and compared to the Interior Penalty (IP) method. A number of numerical experiments are conducted to assess the accuracy and performance of the method. The numerical results and convergence studies indicate that DDG and DDGIC provide accurate solutions and perform well for general flows in thermochemical nonequilibrium.

Diffusion↗
DOE OSTI2026

Integral Kernel Methods for Nonlinear Parabolic-Elliptic Systems

Nonlinear parabolic-elliptic systems arise in many physical, biological, and chemical phenomena such as chemotaxis, ion transport, self-gravitating particles, and Brownian vortices. Existing methods struggle with the strong coupling and high nonlinearity and nonlocality of some of these systems, especially the ill-conditioned, convection-dominated problems. To overcome numerical difficulties, current approaches rely on initial guesses, preconditioning, or iterative techniques with no convergence guarantees. They might suffer from poor scalability, large memory usage, and difficulty to parallelize. Inspired by the connection of parabolic-elliptic systems to stochastic processes, we introduce a novel meshless, monolithic, and fully explicit method that naturally encapsulates the elliptic and parabolic operators into a single step which updates each node deterministically with global information. By being fully quadrature-based, it avoids solving systems of discretized equations and does not utilize initial guesses or preconditioning, while requiring little memory and being easy to parallelize. We first derive the method in an integral kernel formulation with quadratic complexity in the number of integration nodes and then leverage kernel-independent fast multipole methods (FMM) to present a scalable algorithm with linear complexity. We provide numerical examples for the Poisson-Nernst-Planck equations in one, two, and three dimensions, together with the derivation of the integral kernel for each case. Furthermore, the examples demonstrate the fast convergence and scalability of the FMM-accelerated algorithm, as well as its suitability for convection-dominated problems, making it competitive against traditional PDE solvers.

PDE systems↗
DOE OSTI2024

Fast permeability measurement for tight reservoir cores using only initial data of the one chamber pressure pulse decay test

Here, in this study, a mathematical model for fast determination of the permeabilities of tight rocks using measurements taken from the initial period of the One Chamber Pressure Pulse Decay (OC-PPD) test is presented. The model applies to measurements taken both before and after the pressure pulse front has reached the downstream end of the specimen. The analytical solutions for the pressure decay in the upstream chamber are derived based on a parabolic arc approximation of pore pressure distribution along the test specimen. This approximation allows converting the initial–boundary value problem of fluid diffusion in the specimen, governed by partial differential equations, to a system of ordinary differential equations that can be easily solved by explicit formulae. Thus, an explicit formula for the pressure decay rate is obtained, which enables inverse analysis of the initial experimental data to estimate the rock permeability. The proposed method expedites the pulse decay test as it does not require the system to reach equilibrium. The method is validated with three sets of experimental data of the OC-PPD test using helium as the diffusing fluid, for which the relative error of the permeability is found to be less than 6%. This method is particularly useful if the equilibrium time of the pulse decay test for rock specimens with permeabilities in the range of nano-Darcy takes hours or days.

early-time solution↗
DOE OSTI2026

Equation of state for Hf, Ta, W, Re, Os, Ir, Pt, and Au to multi-terapascal pressures from density-functional theory

We present the zero-temperature equation of state (pressure dependence of compression) and phase stability predictions for the 5d-transition metals obtained from all-electron density-functional theory (DFT) calculations. The results compare favorably with experiments but extend beyond current experimental capabilities to 10 TPa. Our study reveals phase changes that are explained from the calculated electronic structure. The cubic face-centered and body-centered structures (fcc and bcc), together with two-, three-, and four-layered hexagonal structures, play major roles under compression. The results’ dependence on the electron exchange and correlation in the DFT approach is investigated, and it is shown that the impact of the choice, while significant at lower pressures, diminishes in the terapascal regime. We further illustrate that the normal parabolic trends in atomic volume and bulk modulus with atomic number, due to the occupation of bonding and anti-bonding 5d states, break down at TPa pressures, suggesting drastically different chemical bonding at these extreme conditions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2026

Explicit Monotone Stable Super-Time-stepping Methods for Finite Time Singularities

We explore a novel way to numerically resolve the scaling behavior of finite-time singularities in solutions of nonlinear parabolic PDEs. The Runge–Kutta–Legendre (RKL) and Runge–Kutta–Gegenbauer (RKG) super-time-stepping methods were originally developed for nonlinear complex physics problems with diffusion. These are multistage single step second-order, forward-in-time methods with no implicit solves. The advantage is that the time-step size for stability scales with stage number 𝑠 as $\mathcal{O}$⁡(𝑠 2 ). Many interesting nonlinear PDEs have finite-time singularities, and the presence of diffusion often limits one to using implicit or semi-implicit time-step methods for stability constraints. Finite-time singularities are particularly challenging due to the large range of scales that one desires to resolve, often with adaptive spatial grids and adaptive time steps. Here, in this study, we show two examples of nonlinear PDEs for which the self-similar singularity structure has time and space scales that are resolvable using the RKL and RKG methods, without forcing even smaller time steps. Compared to commonly used implicit numerical methods, we achieve a significantly smaller run time while maintaining comparable accuracy. We also prove numerical monotonicity for both the RKL and RKG methods under their linear stability conditions for the constant coefficient heat equation, in the case of infinite domain and periodic boundary condition, leading to a theoretical guarantee of the superiority of the RKL and RKG methods over traditional super-time-stepping methods, such as the Runge-Kutta-Chebyshev and the orthogonal Runge-Kutta-Chebyshev methods. Code can be found at https://github.com/ZT220501/SRK-Singularity.

97 MATHEMATICS AND COMPUTING↗