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All sourcesNASA NTRSDOE OSTI
NASA NTRS1998

The Application of Nonstandard Analysis to the Study of Inviscid Shock Wave Jump Conditions

The use of conservation laws in nonconservative form for deriving shock jump conditions by Schwartz distribution theory leads to ambiguous products of generalized functions. Nonstandard analysis is used to define a class of Heaviside functions where the jump from zero to one occurs on an infinitesimal interval. These Heaviside functions differ by their microstructure near x = 0, i.e., by the nature of the rise within the infinitesimal interval it is shown that the conservation laws in nonconservative form can relate the different Heaviside functions used to define jumps in different flow parameters. There are no mathematical or logical ambiguities in the derivation of the jump conditions. An important result is that the microstructure of the Heaviside function of the jump in entropy has a positive peak greater than one within the infinitesimal interval where the jump occurs. This phenomena is known from more sophisticated studies of the structure of shock waves using viscous fluid assumption. However, the present analysis is simpler and more direct.

Farassat, F.↗
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↗
NASA NTRS1992

Essentially nonoscillatory postprocessing filtering methods

High order accurate centered flux approximations used in the computation of numerical solutions to nonlinear partial differential equations produce large oscillations in regions of sharp transitions. Here, we present a new class of filtering methods denoted by Essentially Nonoscillatory Least Squares (ENOLS), which constructs an upgraded filtered solution that is close to the physically correct weak solution of the original evolution equation. Our method relies on the evaluation of a least squares polynomial approximation to oscillatory data using a set of points which is determined via the ENO network. Numerical results are given in one and two space dimensions for both scalar and systems of hyperbolic conservation laws. Computational running time, efficiency, and robustness of method are illustrated in various examples such as Riemann initial data for both Burgers' and Euler's equations of gas dynamics. In all standard cases, the filtered solution appears to converge numerically to the correct solution of the original problem. Some interesting results based on nonstandard central difference schemes, which exactly preserve entropy, and have been recently shown generally not to be weakly convergent to a solution of the conservation law, are also obtained using our filters.

Lafon, F.↗
NASA NTRS1994

Modeling jets in cross flow

Various approaches to the modeling of jets in cross flow are reviewed. These are grouped into four classes, namely: empirical models, integral models, perturbation models, and numerical models. Empirical models depend largely on the correlation of experimental data and are mostly useful for first-order estimates of global properties such as jet trajectory and velocity and temperature decay rates. Integral models are based on some ordinary-differential form of the conservation laws, but require substantial empirical calibration. They allow more details of the flow field to be obtained; simpler versions have to assume similarity of velocity and temperature profiles, but more sophisticated ones can actually calculate these profiles. Perturbation models require little empirical input, but the need for small parameters to ensure convergent expansions limits their application to either the near-field or the far-field. Therefore, they are mostly useful for the study of flow physics. Numerical models are based on conservation laws in partial-differential form. They require little empirical input and have the widest range of applicability. They also require the most computational resources. Although many qualitative and quantitative features of jets in cross flow have been predicted with numerical models, many issues affecting accuracy such as grid resolution and turbulence model are not completely resolved.

Demuren, A. O.↗
NASA NTRS2003

The Effect of the Density Ratio on the Nonlinear Dynamics of the Unstable Fluid Interface

Here we report multiple harmonic theoretical solutions for a complete system of conservation laws, which describe the large-scale coherent dynamics in RTI and RMI for fluids with a finite density ratio in the general three-dimensional case. The analysis yields new properties of the bubble front dynamics. In either RTI or RMI, the obtained dependencies of the bubble velocity and curvature on the density ratio differ qualitatively and quantitatively from those suggested by the models of Sharp (1984), Oron et al. (2001), and Goncharov (2002). We show explicitly that these models violate the conservation laws. For the first time, our theory reveals an important qualitative distinction between the dynamics of the RT and RM bubbles.

Abarzhi, S. I.↗
NASA NTRS2001

Discontinuous Galerkin Methods for NonLinear Differential Systems

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

Barth, Timothy↗
NASA NTRS1990

Some Aspects of Essentially Nonoscillatory (ENO) Formulations for the Euler Equations, Part 3

An essentially nonoscillatory (ENO) formulation is described for hyperbolic systems of conservation laws. ENO approaches are based on smart interpolation to avoid spurious numerical oscillations. ENO schemes are a superset of Total Variation Diminishing (TVD) schemes. In the recent past, TVD formulations were used to construct shock capturing finite difference methods. At extremum points of the solution, TVD schemes automatically reduce to being first-order accurate discretizations locally, while away from extrema they can be constructed to be of higher order accuracy. The new framework helps construct essentially non-oscillatory finite difference methods without recourse to local reductions of accuracy to first order. Thus arbitrarily high orders of accuracy can be obtained. The basic general ideas of the new approach can be specialized in several ways and one specific implementation is described based on: (1) the integral form of the conservation laws; (2) reconstruction based on the primitive functions; (3) extension to multiple dimensions in a tensor product fashion; and (4) Runge-Kutta time integration. The resulting method is fourth-order accurate in time and space and is applicable to uniform Cartesian grids. The construction of such schemes for scalar equations and systems in one and two space dimensions is described along with several examples which illustrate interesting aspects of the new approach.

Chakravarthy, Sukumar R.↗
NASA NTRS2009

The a(4) Scheme-A High Order Neutrally Stable CESE Solver

The CESE development is driven by a belief that a solver should (i) enforce conservation laws in both space and time, and (ii) be built from a nondissipative (i.e., neutrally stable) core scheme so that the numerical dissipation can be controlled effectively. To provide a solid foundation for a systematic CESE development of high order schemes, in this paper we describe a new high order (4-5th order) and neutrally stable CESE solver of a 1D advection equation with a constant advection speed a. The space-time stencil of this two-level explicit scheme is formed by one point at the upper time level and two points at the lower time level. Because it is associated with four independent mesh variables (the numerical analogues of the dependent variable and its first, second, and third-order spatial derivatives) and four equations per mesh point, the new scheme is referred to as the a(4) scheme. As in the case of other similar CESE neutrally stable solvers, the a(4) scheme enforces conservation laws in space-time locally and globally, and it has the basic, forward marching, and backward marching forms. Except for a singular case, these forms are equivalent and satisfy a space-time inversion (STI) invariant property which is shared by the advection equation. Based on the concept of STI invariance, a set of algebraic relations is developed and used to prove the a(4) scheme must be neutrally stable when it is stable. Numerically, it has been established that the scheme is stable if the value of the Courant number is less than 1/3

Chang, Sin-Chung↗
NASA NTRS2007

The alpha(3) Scheme - A Fourth-Order Neutrally Stable CESE Solver

The conservation element and solution element (CESE) development is driven by a belief that a solver should (i) enforce conservation laws in both space and time, and (ii) be built from a non-dissipative (i.e., neutrally stable) core scheme so that the numerical dissipation can be controlled effectively. To provide a solid foundation for a systematic CESE development of high order schemes, in this paper we describe a new 4th-order neutrally stable CESE solver of the advection equation Theta u/Theta + alpha Theta u/Theta x = 0. The space-time stencil of this two-level explicit scheme is formed by one point at the upper time level and three points at the lower time level. Because it is associated with three independent mesh variables u(sup n) (sub j), (u(sub x))(sup n) (sub j) , and (uxz)(sup n) (sub j) (the numerical analogues of u, Theta u/Theta x, and Theta(exp 2)u/Theta x(exp 2), respectively) and four equations per mesh point, the new scheme is referred to as the alpha(3) scheme. As in the case of other similar CESE neutrally stable solvers, the alpha(3) scheme enforces conservation laws in space-time locally and globally, and it has the basic, forward marching, and backward marching forms. These forms are equivalent and satisfy a space-time inversion (STI) invariant property which is shared by the advection equation. Based on the concept of STI invariance, a set of algebraic relations is developed and used to prove that the alpha(3) scheme must be neutrally stable when it is stable. Moreover it is proved rigorously that all three amplification factors of the alpha(3) scheme are of unit magnitude for all phase angles if |v| <= 1/2 (v = alpha delta t/delta x). This theoretical result is consistent with the numerical stability condition |v| <= 1/2. Through numerical experiments, it is established that the alpha(3) scheme generally is (i) 4th-order accurate for the mesh variables u(sup n) (sub j) and (ux)(sup n) (sub j); and 2nd-order accurate for (uxx)(sup n) (sub j). However, in some exceptional cases, the scheme can achieve perfect accuracy aside from round-off errors.

Chang, Sin-Chung↗
NASA NTRS2006

Semi-Analytic Reconstruction of Flux in Finite Volume Formulations

Semi-analytic reconstruction uses the analytic solution to a second-order, steady, ordinary differential equation (ODE) to simultaneously evaluate the convective and diffusive flux at all interfaces of a finite volume formulation. The second-order ODE is itself a linearized approximation to the governing first- and second- order partial differential equation conservation laws. Thus, semi-analytic reconstruction defines a family of formulations for finite volume interface fluxes using analytic solutions to approximating equations. Limiters are not applied in a conventional sense; rather, diffusivity is adjusted in the vicinity of changes in sign of eigenvalues in order to achieve a sufficiently small cell Reynolds number in the analytic formulation across critical points. Several approaches for application of semi-analytic reconstruction for the solution of one-dimensional scalar equations are introduced. Results are compared with exact analytic solutions to Burger s Equation as well as a conventional, upwind discretization using Roe s method. One approach, the end-point wave speed (EPWS) approximation, is further developed for more complex applications. One-dimensional vector equations are tested on a quasi one-dimensional nozzle application. The EPWS algorithm has a more compact difference stencil than Roe s algorithm but reconstruction time is approximately a factor of four larger than for Roe. Though both are second-order accurate schemes, Roe s method approaches a grid converged solution with fewer grid points. Reconstruction of flux in the context of multi-dimensional, vector conservation laws including effects of thermochemical nonequilibrium in the Navier-Stokes equations is developed.

Gnoffo, Peter A.↗
DOE OSTI2023

Operation Optimization using Reinforcement Learning with Integrated Artificial Reasoning Framework

In large and complex systems, operational decision-making requires a systematic analysis with a vast amount of data from both process parameters and component status monitoring. In this paper, we present an integrated artificial reasoning approach for system state transition models that can help operational decision-making with explainable and traceable reasoning. The integrated artificial reasoning framework is a physics-based approach of defining the system structure in a Bayesian network, so we leveraged it in a Markov decision process (MDP) for finding optimal operational solutions. In our proposed framework, the MDP is implemented on a dynamic Bayesian network (DBN), which represents causalities in a system. The multilevel flow modeling was utilized in order to extract these causalities in a more efficient and objective manner. Since multilevel flow modeling is based on the fundamental energy and mass conservation laws, the target system is decomposed into several mass, energy, and information structures, which serve as the basis for a DBN. The MDP consists of the processes of finding a solution for the Bellman equation, which can be derived from the conditional probability equations of the constructed DBN. System operators can capture stochastic system dynamics as multiple subsystem state transitions based on their physical relations and uncertainties coming from the component degradation process or random failures. We analyzed a simplified example system to illustrate finding an optimal operational policy with this approach.

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2021

Explicit structure-preserving geometric particle-in-cell algorithm in curvilinear orthogonal coordinate systems and its applications to whole-device 6D kinetic simulations of tokamak physics

Explicit structure-preserving geometric particle-in-cell (PIC) algorithm in curvilinear orthogonal coordinate systems is developed. The work reported represents a further development of the structure-preserving geometric PIC algorithm achieving the goal of practical applications in magnetic fusion research. The algorithm is constructed by discretizing the field theory for the system of charged particles and electromagnetic field using Whitney forms, discrete exterior calculus, and explicit non-canonical symplectic integration. In addition to the truncated infinitely dimensional symplectic structure, the algorithm preserves exactly many important physical symmetries and conservation laws, such as local energy conservation, gauge symmetry and the corresponding local charge conservation. As a result, the algorithm possesses the long-term accuracy and fidelity required for first-principles-based simulations of the multiscale tokamak physics. The algorithm has been implemented in the SymPIC code, which is designed for high-efficiency massively-parallel PIC simulations in modern clusters. The code has been applied to carry out whole-device 6D kinetic simulation studies of tokamak physics. A self-consistent kinetic steady state for fusion plasma in the tokamak geometry is numerically found with a predominately diagonal and anisotropic pressure tensor. The state also admits a steady-state sub-sonic ion flow in the range of 10 km s -1 , agreeing with experimental observations and analytical calculations Kinetic ballooning instability in the self-consistent kinetic steady state is simulated. It is shown that high-n ballooning modes have larger growth rates than low-n global modes, and in the nonlinear phase the modes saturate approximately in 5 ion transit times at the 2% level by the E × B flow generated by the instability. These results are consistent with early and recent electromagnetic gyrokinetic simulations.

43 PARTICLE ACCELERATORS↗
DOE OSTI2024

Scientific machine learning for closure models in multiscale problems: A review

Here, closure problems are omnipresent when simulating multiscale systems, where some quantities and processes cannot be fully prescribed despite their effects on the simulation's accuracy. Recently, scientific machine learning approaches have been proposed as a way to tackle the closure problem, combining traditional (physics-based) modeling with data-driven (machine-learned) techniques, typically through enriching differential equations with neural networks. This paper reviews the different reduced model forms, distinguished by the degree to which they include known physics, and the different objectives of a priori and a posteriori learning. The importance of adhering to physical laws (such as symmetries and conservation laws) in choosing the reduced model form and choosing the learning method is discussed. The effect of spatial and temporal discretization and recent trends toward discretization-invariant models are reviewed. In addition, we make the connections between closure problems and several other research disciplines: inverse problems, Mori-Zwanzig theory, and multi-fidelity methods. In conclusion, much progress has been made with scientific machine learning approaches for solving closure problems, but many challenges remain. In particular, the generalizability and interpretability of learned models is a major issue that needs to be addressed further.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2000

Parallel CE/SE Computations via Domain Decomposition

This paper describes the parallelization strategy and achieved parallel efficiency of an explicit time-marching algorithm for solving conservation laws. The Space-Time Conservation Element and Solution Element (CE/SE) algorithm for solving the 2D and 3D Euler equations is parallelized with the aid of domain decomposition. The parallel efficiency of the resultant algorithm on a Silicon Graphics Origin 2000 parallel computer is checked.

Himansu, Ananda↗
DOE OSTI2026

A sweeping positivity-preserving high-order finite difference WENO scheme for Euler equations

We develop a simple, high-order, conservative and robust positivity-preserving sweeping procedure for the density and the nonlinear pressure function in the compressible Euler equations. Using the scaling limiter in Zhang and Shu (J Comput Phys 229:3091–3120, 2010), we obtain a non-trivial extension of the scalar sweeping technique in Liu et al. (J Sci Comput 73:1028–1071, 2017) for the positivity of pressure. The sweeping procedure developed in this paper is a post-processing technique, which can be applied to any concave functions of the conserved variables in hyperbolic conservation law systems. Thus, it has applications beyond the Euler equations. This procedure preserves positivity and conservation of physical quantities without destroying the accuracy of the underlying scheme. The algorithm works for general schemes including finite difference, finite volume and discontinuous Galerkin methods; however, in this paper we focus on finite difference weighted essentially non-oscillatory (WENO) methods. As a result, we provide numerical tests of the fifth-order finite difference WENO scheme to demonstrate the accuracy and robustness of the technique.

Compressible Euler equations↗
DOE OSTI2021

Angular momentum effects in fission

Background: The role of angular momentum in fission has long been discussed but the observable effects are difficult to quantify. Purpose: Here we discuss a variety of effects associated with angular momentum in fission and present quantitative illustrations. Methods: We employ the fission simulation model freya , which is well suited for this purpose because it obeys all conservation laws, including linear and angular momentum conservation at each step of the process. We first discuss the implementation of angular momentum in freya and then assess particular observables, including various correlated observables. We also study potential effects of neutron-induced fission of the low-lying isomeric state of U 235 relative to the ground state. Results: The fluctuations inherent in the fission process ensure that the spin of the initial compound nucleus has only a small influence on the fragment spins, which are therefore nearly uncorrelated. There is a marked correlation between the spin magnitude of the fission fragments and the photon multiplicity. We also consider the dynamical anisotropy caused by the rotation of an evaporating fragment and study especially the distribution of the projected neutron-neutron opening angles, showing that while it is dominated by the effect of the evaporation recoils, it is possible to extract the signal of the dynamical anisotropy by means of a Fourier decomposition. Finally, we note that the use of an isomeric target, U 235 m ( n th ,f), may enhance the symmetric yields and can thus result in higher neutron multiplicities for low total fragment kinetic energies. Conclusions: While the initial angular momentum of the fissioning nucleus tends to have little effect on the observables, those of the produced fragments influence the emitted neutrons and photons in a significant and correlated manner which may be exploited experimentally to elucidate the fission process.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS2012

Development of a Robust and Efficient Parallel Solver for Unsteady Turbomachinery Flows

The traditional design and analysis practice for advanced propulsion systems relies heavily on expensive full-scale prototype development and testing. Over the past decade, use of high-fidelity analysis and design tools such as CFD early in the product development cycle has been identified as one way to alleviate testing costs and to develop these devices better, faster and cheaper. In the design of advanced propulsion systems, CFD plays a major role in defining the required performance over the entire flight regime, as well as in testing the sensitivity of the design to the different modes of operation. Increased emphasis is being placed on developing and applying CFD models to simulate the flow field environments and performance of advanced propulsion systems. This necessitates the development of next generation computational tools which can be used effectively and reliably in a design environment. The turbomachinery simulation capability presented here is being developed in a computational tool called Loci-STREAM [1]. It integrates proven numerical methods for generalized grids and state-of-the-art physical models in a novel rule-based programming framework called Loci [2] which allows: (a) seamless integration of multidisciplinary physics in a unified manner, and (b) automatic handling of massively parallel computing. The objective is to be able to routinely simulate problems involving complex geometries requiring large unstructured grids and complex multidisciplinary physics. An immediate application of interest is simulation of unsteady flows in rocket turbopumps, particularly in cryogenic liquid rocket engines. The key components of the overall methodology presented in this paper are the following: (a) high fidelity unsteady simulation capability based on Detached Eddy Simulation (DES) in conjunction with second-order temporal discretization, (b) compliance with Geometric Conservation Law (GCL) in order to maintain conservative property on moving meshes for second-order time-stepping scheme, (c) a novel cloud-of-points interpolation method (based on a fast parallel kd-tree search algorithm) for interfaces between turbomachinery components in relative motion which is demonstrated to be highly scalable, and (d) demonstrated accuracy and parallel scalability on large grids (approx 250 million cells) in full turbomachinery geometries.

West, Jeff↗
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↗