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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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All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

An Entropy-Based Approach to Nonlinear Stability

Many numerical methods used in computational fluid dynamics (CFD) incorporate an artificial dissipation term to suppress spurious oscillations and control nonlinear instabilities. The same effect can be accomplished by using upwind techniques, sometimes augmented with limiters to form Total Variation Diminishing (TVD) schemes. An analysis based on numerical satisfaction of the second law of thermodynamics allows many such methods to be compared and improved upon. A nonlinear stability proof is given for discrete scalar equations arising from a conservation law. Solutions to such equations are bounded in the L sub 2 norm if the second law of thermodynamics is satisfied in a global sense over a periodic domain. It is conjectured that an analogous statement is true for discrete equations arising from systems of conservation laws. Analysis and numerical experiments suggest that a more restrictive condition, a positive entropy production rate in each cell, is sufficient to exclude unphysical phenomena such as oscillations and expansion shocks. Construction of schemes which satisfy this condition is demonstrated for linear and nonlinear wave equations and for the one-dimensional Euler equations.

Merriam, Marshal L.↗
NASA NTRS1993

A priori error estimates for an hp-version of the discontinuous Galerkin method for hyperbolic conservation laws

A priori error estimates are derived for hp-versions of the finite element method for discontinuous Galerkin approximations of a model class of linear, scalar, first-order hyperbolic conservation laws. These estimates are derived in a mesh dependent norm in which the coefficients depend upon both the local mesh size h(sub K) and a number p(sub k) which can be identified with the spectral order of the local approximations over each element.

Bey, Kim S.↗
DOE OSTI2026

Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs

Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes, specifically for problems where the data obey a conservation law arising from an underlying partial differential equation. Our approach combines discrete exterior calculus and nonlinear optimal recovery to infer relationships between vertex and edge values. This framework yields data-driven predictions with uncertainty quantification across the entire graph, even when observations are limited to a subset of vertices and edges. By minimizing the reproducing kernel Hilbert space norm while penalizing kernel complexity through maximum likelihood estimation, our method ensures that the resulting surrogate strictly enforces conservation laws without overfitting. We demonstrate our method on two representative applications: subsurface flow in fracture networks and arterial blood flow. Finally, the results demonstrate that the method maintains high accuracy and well-calibrated uncertainty estimates even under severe data scarcity, highlighting its potential for scientific applications where limited data and reliable uncertainty quantification are critical.

Dirichlet-to-Neumann map↗
DOE OSTI2023

Probing the chiral magnetic wave with charge-dependent flow measurements in Pb-Pb collisions at the LHC

The Chiral Magnetic Wave (CMW) phenomenon is essential to provide insights into the strong interaction in QCD, the properties of the quark-gluon plasma, and the topological characteristics of the early universe, offering a deeper understanding of fundamental physics in high-energy collisions. Measurements of the charge-dependent anisotropic flow coefficients are studied in Pb-Pb collisions at center-of-mass energy per nucleon-nucleon collision $\sqrt{s_{\textrm{NN}}}$ = 5.02 TeV to probe the CMW. In particular, the slope of the normalized difference in elliptic (v 2 ) and triangular (v 3 ) flow coefficients of positively and negatively charged particles as a function of their event-wise normalized number difference, is reported for inclusive and identified particles. The slope ${r}_3^{\textrm{Norm}}$ is found to be larger than zero and to have a magnitude similar to ${r}_2^{\textrm{Norm}}$, thus pointing to a large background contribution for these measurements. Furthermore, ${r}_2^{\textrm{Norm}}$ can be described by a blast wave model calculation that incorporates local charge conservation. In addition, using the event shape engineering technique yields a fraction of CMW (f CMW ) contribution to this measurement which is compatible with zero. This measurement provides the very first upper limit for f CMW , and in the 10–60% centrality interval it is found to be 26% (38%) at 95% (99.7%) confidence level.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS1995

An hp-adaptivity and error estimation for hyperbolic conservation laws

This paper presents an hp-adaptive discontinuous Galerkin method for linear hyperbolic conservation laws. A priori and a posteriori error estimates are derived in mesh-dependent norms which reflect the dependence of the approximate solution on the element size (h) and the degree (p) of the local polynomial approximation. The a posteriori error estimate, based on the element residual method, provides bounds on the actual global error in the approximate solution. The adaptive strategy is designed to deliver an approximate solution with the specified level of error in three steps. The a posteriori estimate is used to assess the accuracy of a given approximate solution and the a priori estimate is used to predict the mesh refinements and polynomial enrichment needed to deliver the desired solution. Numerical examples demonstrate the reliability of the a posteriori error estimates and the effectiveness of the hp-adaptive strategy.

Bey, Kim S.↗
NASA NTRS2003

Time as an Observable in Nonrelativistic Quantum Mechanics

The argument follows from the viewpoint that quantum mechanics is taken not in the usual form involving vectors and linear operators in Hilbert spaces, but as a boundary value problem for a special class of partial differential equations-in the present work, the nonrelativistic Schrodinger equation for motion of a structureless particle in four- dimensional space-time in the presence of a potential energy distribution that can be time-as well as space-dependent. The domain of interest is taken to be one of two semi-infinite boxes, one bounded by two t=constant planes and the other by two t=constant planes. Each gives rise to a characteristic boundary value problem: one in which the initial, input values on one t=constant wall are given, with zero asymptotic wavefunction values in all spatial directions, the output being the values on the second t=constant wall; the second with certain input values given on both z=constant walls, with zero asymptotic values in all directions involving time and the other spatial coordinates, the output being the complementary values on the z=constant walls. The first problem corresponds to ordinary quantum mechanics; the second, to a fully time-dependent version of a problem normally considered only for the steady state (time-independent Schrodinger equation). The second problem is formulated in detail. A conserved indefinite metric is associated with space-like propagation, where the sign of the norm of a unidirectional state corresponds to its spatial direction of travel.

Hahne, G. E.↗
DOE OSTI2023

Cooking methods and kitchen ventilation availability, usage, perceived performance and potential in Canadian homes

Cooking is a substantial contributor to air pollutant exposures in many residences. Effective use of kitchen ventilation can mitigate exposure; however, information on its availability, usage, and potential to increase its use across the population has been limited. This study aimed to obtain nationally representative information on cooking methods, kitchen ventilation availability and usage, and the potential for education to increase effective usage. An online survey was sent to a representative sample of Canadian homes to collect data on cooking methods, the presence and use of mechanical kitchen ventilation devices, perceived device performance, and willingness to implement mitigation strategies. Responses were weighted to match key demographic factors and analyzed using non-parametric statistics. Among the 4500 respondents, 90% had mechanical ventilation devices over the cooktop (66% of which were vented to the outside), and 30% reported regularly using their devices. Devices were used most often for deep-frying, followed by stir-frying, sautéing or pan-frying, indoor grilling, boiling or steaming. Almost half reported rarely or never using their ventilation devices during baking or oven self-cleaning. Only 10% were fully satisfied with their devices. More frequent use was associated with the device being vented to the outdoors, having more than two speed settings, quiet operation if only one speed, covering over half of the cooktop, and higher perceived effectiveness. After being informed of the benefits of kitchen ventilation, 64% indicated they would consider using their devices more often, preferentially using back burners with ventilation, and/or using higher ventilation device settings when needed. This study provides population-representative data on the most used cooking methods, kitchen ventilation availability and usage, and influencing factors in Canadian homes. Such data are needed for exposure assessments and evaluating the potential to mitigate cooking-related pollutant exposures via more effective use of kitchen ventilation. The data can be reasonably extrapolated to the United States, given the similarities in residential construction practices and cultural norms between the two countries.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
NASA NTRS2019

Entropy Stable Method for the Euler Equations Revisited: Central Differencing via Entropy Splitting and SBP

The two decades old high order central differencing via entropy splitting and summation-by-parts (SBP) difference boundary closure of Ols- son & Oliger (1994), Gerritsen & Olsson (1996), and Yee et al. (2000) is revisited. The entropy splitting is a form of skew-symmetric splitting of the nonlinear Euler flux derivatives. Central differencing applied to the entropy splitting form of the Euler flux derivatives together with SBP difference operators will, hereafter, be referred to as entropy split schemes. This study is prompted by the recent growing interest in numerical methods for which a discrete entropy conservation law holds, a discrete global entropy conservation can be proved and/or the numerical method possesses a stable entropy in the framework of SBP difference operators and L2-energy norm estimate. The objective of this paper is to recast the entropy split scheme as the re- cent definition of an entropy stable method for central differencing with SBP operators for both periodic and non-periodic boundary conditions for non- linear Euler equations. Standard high order spatial central differencing as well as high order central spatial DRP (dispersion relation preserving) spatial differencing is part of the entropy stable methodology framework. Long time integration of 2D and 3D test cases is included to show the comparison of this efficient entropy stable method with the Tadmor-type of entropy conservative methods. Studies also include the comparison among the three skew-symmetric splittings on their nonlinear stability and accuracy performance without added numerical dissipations for smooth flows. These are, namely, entropy splitting, Ducros et al. splitting and the Kennedy & Grub- ber splitting.

Sjogreen, Bjorn↗
NASA NTRS1990

Least-squares finite element methods for compressible Euler equations

A method based on backward finite differencing in time and a least-squares finite element scheme for first-order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the L-sq-norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high-order elements, computed solutions based on the L-sq method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted H1-norm of the residual is proposed and leads to a successful scheme with high-degree elements. Finally, a conservative least-squares finite element method is also developed. Numerical results for two-dimensional problems are given to demonstrate the shock resolution of the methods and compare different approaches.

Jiang, Bo-Nan↗
NASA NTRS1992

Nonlinearly stable compact schemes for shock calculations

The applications of high-order, compact finite difference methods in shock calculations are discussed. The main concern is to define a local mean which will serve as a reference for introducing a local nonlinear limiting to control spurious numerical oscillations while maintaining the formal accuracy of the scheme. For scalar conservation laws, the resulting schemes can be proven total-variation stable in one space dimension and maximum-norm stable in multiple space dimensions. Numerical examples are shown to verify accuracy and stability of such schemes for problems containing shocks. These ideas can also be applied to other implicit schemes such as the continuous Galerkin finite element methods.

Cockburn, Bernardo↗
NASA NTRSDate not supplied

A New Approach for a Wider Class of Entropy Split Methods for Compressible Gas Dynamics and MHD

The high order entropy split methods of Sjögreen & Yee [1, 2] by entropy splitting of the compressible Euler (inviscid) flux derivatives for a thermally-perfect gas are based on Harten’s entropy function [3, 4, 5]. Their derivation takes advantage of the homogeneity property of Euler flux, symmetrizable Euler flux derivatives and energy-norm stability in conjunction with high order classical spatial central, DRP (dispersion relation-preserving) [6, 7, 8] or Padé (compact) spatial discretizations [9] with summation-by-parts (SBP) operators [10]. Our entropy split methods have been proven entropy conserving and stable [1, 11, 12]. Our proofs do not rely on a two-point numerical flux, but rather only a linear difference operator is required to derive these methods. To extend the entropy split method for the MHD, we used the Godunov symmetrizable non-conservative MHD form [12, 13, 14]. These high order entropy split methods not only preserve certain physical properties of the chosen governing equations but are also known to either improve numerical stability, and/or minimize aliasing errors in long time integration of turbulent flow computations without the aid of added numerical dissipation. In our previous published work, extensive error norm comparison with grid refinement was performed to show the high accuracy performance of these methods. These studies also showed how well the entropy split methods conserve the entropy, momentum and mass, and preserve the kinetic energy for long time integration of the various flows [1, 2, 12, 13, 14]. The objective of the present work is to use a new approach to obtain a wider class of entropy split methods consisting of a two-point numerical flux portion and a non-conservative portion in such a way that the homogeneity property of the compressible Euler flux is not required. For high order classical spatial central, DRP (dispersion relation-preserving) or Padé (compact) spatial discretizations, this new approach can be proven to be entropy conservative with conservative spatial dsicretizations while at the same time allowing a wider class of symmetrizable inviscid flux derivatives. We also use this generalization to derive an entropy split scheme that is entropy conserving for the equations of MHD without the homogeneity property using the Godunov symmetrizable ideal MHD formulation [15].

High Order Physical Preserving Methods↗
NASA NTRS1979

Stability bounds for the control of large space structures

Balas (1977) has discussed the stability problem of reduced-order regulators and estimators in terms of control and observation 'spillover'. The term 'control spillover' was used to define that part of the feedback control which excites the uncontrolled (or residual) modes, and 'observation spillover' was used to define that part of the measurement which is contaminated by residual modes. In this paper, two sufficient conditions are derived via Lyapunov methods for asymptotic stability of large space structures using a class of reduced-order controllers. These conditions give allowable bounds on the spectral norms of control and observation spillover terms. The sufficient condition given by a specified inequality equation appears to be less conservative, and should be useful as a design tool for the control of large space structures.

Joshi, S. M.↗
NASA NTRS1987

A Taylor weak-statement algorithm for hyperbolic conservation laws

Finite element analysis, applied to computational fluid dynamics (CFD) problem classes, presents a formal procedure for establishing the ingredients of a discrete approximation numerical solution algorithm. A classical Galerkin weak-statement formulation, formed on a Taylor series extension of the conservation law system, is developed herein that embeds a set of parameters eligible for constraint according to specification of suitable norms. The derived family of Taylor weak statements is shown to contain, as special cases, over one dozen independently derived CFD algorithms published over the past several decades for the high speed flow problem class. A theoretical analysis is completed that facilitates direct qualitative comparisons. Numerical results for definitive linear and nonlinear test problems permit direct quantitative performance comparisons.

Baker, A. J.↗
NASA NTRS1991

Time domain convergence properties of Lyapunov stable penalty methods

Linear hyperbolic partial differential equations are analyzed using standard techniques to show that a sequence of solutions generated by the Liapunov stable penalty equations approaches the solution of the differential-algebraic equations governing the dynamics of multibody problems arising in linear vibrations. The analysis does not require that the system be conservative and does not impose any specific integration scheme. Variational statements are derived which bound the error in approximation by the norm of the constraint violation obtained in the approximate solutions.

Kurdila, A. J.↗
NASA NTRS2011

Discretely Conservative Finite-Difference Formulations for Nonlinear Conservation Laws in Split Form: Theory and Boundary Conditions

Simulations of nonlinear conservation laws that admit discontinuous solutions are typically restricted to discretizations of equations that are explicitly written in divergence form. This restriction is, however, unnecessary. Herein, linear combinations of divergence and product rule forms that have been discretized using diagonal-norm skew-symmetric summation-by-parts (SBP) operators, are shown to satisfy the sufficient conditions of the Lax-Wendroff theorem and thus are appropriate for simulations of discontinuous physical phenomena. Furthermore, special treatments are not required at the points that are near physical boundaries (i.e., discrete conservation is achieved throughout the entire computational domain, including the boundaries). Examples are presented of a fourth-order, SBP finite-difference operator with second-order boundary closures. Sixth- and eighth-order constructions are derived, and included in E. Narrow-stencil difference operators for linear viscous terms are also derived; these guarantee the conservative form of the combined operator.

Fisher, Travis C.↗
DOE OSTI2026

A variational mimetic finite difference method for elliptic interface problems on non-matching polytopal meshes with geometric interface inconsistencies

A new variational mimetic finite difference method for elliptic interface problems with perfect and imperfect thermal contacts on non-matching polytopal meshes with geometric interface inconsistencies is developed and analyzed theoretically and numerically. The method is defined on multiple non-matching submeshes with gaps and overlaps along their interfaces. The discrete equations are derived from a minimization problem for the augmented Dirichlet functional. For a perfect thermal contact, the functional uses a modified mimetic gradient with extended stencil which couples unknowns from both sides of an interface, as well as penalty terms to enforce weak continuity of temperature across the interface. The method leads to a symmetric positive definite matrix for any scaling of the penalty terms. For an imperfect thermal contact, the Dirichlet functional is supplemented with a quadratic jump term along the interface related to the interface thermal resistance. We prove that the method conserves the total heat flux across each interface. In conclusion, the obtained results are verified with numerical experiments showing convergence in the discrete L 2 and L ∞ norms.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Quantum field theory, worldline theory, and spin magnitude change in orbital evolution

A previous paper [Z. Bern et al., Binary dynamics through the fifth power of spin at 𝑂⁡(𝐺 2 ), Phys. Rev. Lett. 130, 201402 (2023)] identified a puzzle stemming from the amplitudes-based approach to spinning bodies in general relativity: additional Wilson coefficients appear compared to current worldline approaches to conservative dynamics of generic astrophysical objects, including neutron stars. In this paper we clarify the nature of analogous Wilson coefficients in the simpler theory of electrodynamics. We analyze the original field-theory construction, identifying definite-spin states some of which have negative norms, and relating the additional Wilson coefficients in the classical theory to transitions between different quantum spin states. We produce a new version of the theory which also has additional Wilson coefficients, but no negative-norm states. We match, through 𝒪⁡(𝛼 2 ) and 𝒪⁡(𝑆 2 ), the Compton amplitudes of these field theories with those of a modified worldline theory with extra degrees of freedom introduced by releasing the spin supplementary condition. We build an effective two-body Hamiltonian that matches the impulse and spin kick of the modified field theory and of the worldline theory, displaying additional Wilson coefficients compared to standard worldline approaches. The results are then compactly expressed in terms of an eikonal formula. Our key conclusion is that, contrary to standard approaches, while the magnitude of the spin tensor is still conserved, the magnitude of the spin vector can change under conserved Hamiltonian dynamics and this change is governed by the additional Wilson coefficients. For specific values of Wilson coefficients the results are equivalent to those from a definite spin obeying the spin supplementary condition, but for generic values they are physically inequivalent. These results warrant detailed studies of the corresponding issues in general relativity.

classical black holes↗
NASA NTRS2013

High-Order Entropy Stable Formulations for Computational Fluid Dynamics

A systematic approach is presented for developing entropy stable (SS) formulations of any order for the Navier-Stokes equations. These SS formulations discretely conserve mass, momentum, energy and satisfy a mathematical entropy inequality. They are valid for smooth as well as discontinuous flows provided sufficient dissipation is added at shocks and discontinuities. Entropy stable formulations exist for all diagonal norm, summation-by-parts (SBP) operators, including all centered finite-difference operators, Legendre collocation finite-element operators, and certain finite-volume operators. Examples are presented using various entropy stable formulations that demonstrate the current state-of-the-art of these schemes.

Carpenter, Mark H.↗