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All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Tau approximation techniques for identification of coefficients in parabolic PDE

A variant of the Tau method, called the weak Tau method, is developed on the basis of the weak form of the PDE for use in least-squares parameter estimation; also presented is a suitable abstract convergence framework. The emphasis is on the theoretical framework that allows treatment of the weak Tau method when it is applied to a wide class of inverse problems, including those for diffusion-advection equations, the Fokker-Planck model for population dynamics, and damped beam equations. Extensive numerical testing of the weak Tau method has demonstrated that it compares quite favorably with existing methods.

Banks, H. T.↗
NASA NTRS1991

Progress with multigrid schemes for hypersonic flow problems

Several multigrid schemes are considered for the numerical computation of viscous hypersonic flows. For each scheme, the basic solution algorithm uses upwind spatial discretization with explicit multistage time stepping. Two level versions of the various multigrid algorithms are applied to the two dimensional advection equation, and Fourier analysis is used to determine their damping properties. The capabilities of the multigrid methods are assessed by solving three different hypersonic flow problems. Some new multigrid schemes based on semicoarsening strategies are shown to be quite effective in relieving the stiffness caused by the high aspect ratio cells required to resolve high Reynolds number flows. These schemes exhibit good convergence rates for Reynolds numbers up to 200 x 10(exp 6) and Mach numbers up to 25.

Radespiel, R.↗
NASA NTRS1993

Multigrid schemes for viscous hypersonic flows

Several multigrid schemes are considered for the numerical computation of viscous hypersonic flows. For each scheme, the basic solution algorithm employs upwind spatial discretization with explicit multistage time stepping. Two-level versions of the various multigrid algorithms are applied to the two-dimensional advection equation, and Fourier analysis is used to determine their damping properties. The capabilities of the multigrid methods are assessed by solving two different hypersonic flow problems. Some new multigrid schemes, based on semicoarsening strategies, are shown to be quite effective in relieving the stiffness caused by the high-aspect-ratio cells required to resolve high Reynolds number flows. These schemes exhibit good convergence rates for Reynolds numbers up to 200 x 10(exp 6).

Swanson, R. C.↗
NASA NTRS1995

Time-dependent simulation of oblique MHD cosmic-ray shocks using the two-fluid model

Using a new, second-order accurate numerical method we present dynamical simulations of oblique MHD cosmic-ray (CR)-modified plane shock evolution. Most of the calculations are done with a two-fluid model for diffusive shock acceleration, but we provide also comparisons between a typical shock computed that way against calculations carried out using the more complete, momentum-dependent, diffusion-advection equation. We also illustrate a test showing that these simulations evolve to dynamical equilibria consistent with previously published steady state analytic calculations for such shocks. In order to improve understanding of the dynamical role of magnetic fields in shocks modified by CR pressure we have explored for time asymptotic states the parameter space of upstream fast mode Mach number, M(sub f), and plasma beta. We compile the results into maps of dynamical steady state CR acceleration efficiency, epsilon(sub c). We have run simulations using constant, and nonisotropic, obliquity (and hence spatially) dependent forms of the diffusion coefficient kappa. Comparison of the results shows that while the final steady states achieved are the same in each case, the history of CR-MHD shocks can be strongly modified by variations in kappa and, therefore, in the acceleration timescale. Also, the coupling of CR and MHD in low beta, oblique shocks substantially influences the transient density spike that forms in strongly CR-modified shocks. We find that inside the density spike a MHD slow mode wave can be generated that eventually steepens into a shock. A strong layer develops within the density spike, driven by MHD stresses. We conjecture that currents in the shear layer could, in nonplanar flows, results in enhanced particle accretion through drift acceleration.

Frank, Adam↗
NASA NTRS2006

On Space-Time Inversion Invariance and its Relation to Non-Dissipatedness of a CESE Core Scheme

The core motivating ideas of the space-time CESE method are clearly presented and critically analyzed. It is explained why these ideas result in all the simplifying and enabling features of the CESE method. A thorough discussion of the a scheme, a two-level non-dissipative CESE solver of a simple advection equation with two independent mesh variables and two equations per mesh point is also presented. It is shown that the scheme possesses some rather intriguing properties such as: (i) its two independent mesh variables separately satisfy two decoupled three-level leapfrog schemes and (ii) it shares with the leapfrog scheme the same amplification factors, even though the a scheme and the leapfrog scheme have completely different origins and structures. It is also explained why the leapfrog scheme is not as robust as the a scheme. The amplification factors/matrices of several non-dissipative schemes are carefully studied and the key properties that contribute to their non-dissipatedness are clearly spelled out. Finally we define and establish space-time inversion (STI) invariance for several non-dissipative schemes and show that their non-dissipatedness is a result of their STI invariance.

Chang, Sin-Chung↗
NASA NTRS2008

The a(3) Scheme--A Fourth-Order Space-Time Flux-Conserving and 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 non-dissipative (i.e., neutrally stable) core scheme so that the numerical dissipation can be controlled effectively. To initiate a systematic CESE development of high order schemes, in this paper we provide a thorough discussion on the structure, consistency, stability, phase error, and accuracy of a new 4th-order space-time flux-conserving and neutrally stable CESE solver of an 1D scalar advection equation. 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 (the numerical analogues of the dependent variable and its 1st-order and 2ndorder spatial derivatives, respectively) and three equations per mesh point, the new scheme is referred to as the a(3) scheme. Through the von Neumann analysis, it is shown that the a(3) scheme is stable if and only if the Courant number is less than 0.5. Moreover, it is established numerically that the a(3) scheme is 4th-order accurate.

Chang, Sin-Chung↗
NASA NTRS2000

Low-dimensional Representation of Error Covariance

Ensemble and reduced-rank approaches to prediction and assimilation rely on low-dimensional approximations of the estimation error covariances. Here stability properties of the forecast/analysis cycle for linear, time-independent systems are used to identify factors that cause the steady-state analysis error covariance to admit a low-dimensional representation. A useful measure of forecast/analysis cycle stability is the bound matrix, a function of the dynamics, observation operator and assimilation method. Upper and lower estimates for the steady-state analysis error covariance matrix eigenvalues are derived from the bound matrix. The estimates generalize to time-dependent systems. If much of the steady-state analysis error variance is due to a few dominant modes, the leading eigenvectors of the bound matrix approximate those of the steady-state analysis error covariance matrix. The analytical results are illustrated in two numerical examples where the Kalman filter is carried to steady state. The first example uses the dynamics of a generalized advection equation exhibiting nonmodal transient growth. Failure to observe growing modes leads to increased steady-state analysis error variances. Leading eigenvectors of the steady-state analysis error covariance matrix are well approximated by leading eigenvectors of the bound matrix. The second example uses the dynamics of a damped baroclinic wave model. The leading eigenvectors of a lowest-order approximation of the bound matrix are shown to approximate well the leading eigenvectors of the steady-state analysis error covariance matrix.

Tippett, Michael K.↗
NASA NTRS2012

Anomaly Detection in Test Equipment via Sliding Mode Observers

Nonlinear observers were originally developed based on the ideas of variable structure control, and for the purpose of detecting disturbances in complex systems. In this anomaly detection application, these observers were designed for estimating the distributed state of fluid flow in a pipe described by a class of advection equations. The observer algorithm uses collected data in a piping system to estimate the distributed system state (pressure and velocity along a pipe containing liquid gas propellant flow) using only boundary measurements. These estimates are then used to further estimate and localize possible anomalies such as leaks or foreign objects, and instrumentation metering problems such as incorrect flow meter orifice plate size. The observer algorithm has the following parts: a mathematical model of the fluid flow, observer control algorithm, and an anomaly identification algorithm. The main functional operation of the algorithm is in creating the sliding mode in the observer system implemented as software. Once the sliding mode starts in the system, the equivalent value of the discontinuous function in sliding mode can be obtained by filtering out the high-frequency chattering component. In control theory, "observers" are dynamic algorithms for the online estimation of the current state of a dynamic system by measurements of an output of the system. Classical linear observers can provide optimal estimates of a system state in case of uncertainty modeled by white noise. For nonlinear cases, the theory of nonlinear observers has been developed and its success is mainly due to the sliding mode approach. Using the mathematical theory of variable structure systems with sliding modes, the observer algorithm is designed in such a way that it steers the output of the model to the output of the system obtained via a variety of sensors, in spite of possible mismatches between the assumed model and actual system. The unique properties of sliding mode control allow not only control of the model internal states to the states of the real-life system, but also identification of the disturbance or anomaly that may occur.

Solano, Wanda M.↗
DOE OSTI2023

1-D coupled surface flow and transport equations revisited via the physics-informed neural network approach

The de Saint-Venant equation (SVE) and advection–diffusion equation (ADE) are commonly employed to solve solute transport problems in surface water. In this work, we propose a mesh-free method based on the physics-informed neural network (PINN) to solve the one dimensional (1-D) SVE, ADE, and the coupled SVE and ADE (SVE-ADE) under various initial and boundary conditions. The PINN model extends the architecture of deep neural networks (DNNs) with implementation of loss function, which are additionally subject to constraints imposed by the physical laws of SVE and ADE, along with their initial and boundary conditions. In such a manner, PINNs can be quickly steered to the true solution while obeying the physical laws. The results of PINN model are compared with the analytical and/or numerical solutions under various conditions to investigate its accuracy and efficiency in solving the SVE, ADE, and SVE-ADE. Our results indicate PINN can accurately simulate the shock wave morphology and avoid numerical dissipation in unsteady flow condition. The PINN method outweighed traditional numerical methods in several aspects, including its ability to function with small amounts of data, no grid discretization, and random selection of sampling points, etc. Additionally, the PINN method is also suitable for solving inverse problems with sparse and noisy data. With 1% noise and 2000 initial and boundary condition points (N u ), the errors of the estimated flow rate (v) and diffusion coefficient (D) are 0.003% and 0.105%, respectively, which indicate the accuracy and robustness of the proposed method. Finally, our results indicate the capability and robustness of the proposed PINN methodology for solving multi-physics problems, irrespective of the presence of sparse and noisy data in the training dataset.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

Understanding latent timescales in neural ordinary differential equation models of advection-dominated dynamical systems

The neural ordinary differential equation (ODE) framework has shown considerable promise in recent years in developing highly accelerated surrogate models for complex physical systems characterized by partial differential equations (PDEs). For PDE-based systems, state-of-the-art neural ODE strategies leverage a two-step procedure to achieve this acceleration: a nonlinear dimensionality reduction step provided by an autoencoder, and a time integration step provided by a neural-network based model for the resultant latent space dynamics (the neural ODE). This work explores the applicability of such autoencoder-based neural ODE strategies for PDEs in which advection terms play a critical role. More specifically, alongside predictive demonstrations, physical insight into the sources of model acceleration (i.e., how the neural ODE achieves its acceleration) is the scope of the current study. Such investigations are performed by quantifying the effects of both autoencoder and neural ODE components on latent system time-scales using eigenvalue analysis of dynamical system Jacobians. To this end, the sensitivity of various critical training parameters – de-coupled versus end-to-end training, latent space dimensionality, and the role of training trajectory length, for example – to both model accuracy and the discovered latent system timescales is quantified. Furthermore, this work specifically uncovers the key role played by the training trajectory length (the number of rollout steps in the loss function during training) on the latent system timescales: larger trajectory lengths correlate with an increase in limiting neural ODE time-scales, and optimal neural ODEs are found to recover the largest time-scales of the full-order (ground-truth) system. Demonstrations are performed across fundamentally different unsteady fluid dynamics configurations influenced by advection: (1) the Kuramoto–Sivashinsky equations (2) Hydrogen-Air channel detonations (the compressible reacting Navier–Stokes equations with detailed chemistry), and (3) 2D Atmospheric flow.

Advection-dominated dynamical systems↗
DOE OSTI2020

Vertical Transport of Water/Contaminant Due to Intermittent Water Influx on the Ground Surface of Vadose Zone - 20088

For movement of moisture and transport of contaminants in the vertical direction, two equations, the Richards equation and the advection-dispersion equation, are solved in a vadose zone for prescribed water and contaminant influx conditions at the ground surface. Two different boundary conditions are considered, one with a single peak of sinusoidal variation and four peaks of finite duration, each of which is sinusoidal in shape. The moisture migration is solved and then the contaminant transport, which is coupled to moisture movement, is solved. In the case of a single influx on the ground surface, at first the hydraulic head and moisture content show a peak. Then above the peak and below the ground surface, the peaks tend to die out, showing a tendency to return to the initial state. The vertical profile of concentration also shows a single peak, which soon becomes stagnant because downward water migration is faster than concentration profile movement. In the case of four intermittent array of peaks, the hydraulic head and moisture content profiles tend to merge and combine together. The concentration profiles show appearance of four peaks clearly. However, although the peak value of concentration remains quite constant, the values may vary with the sorption coefficient because of repeated influx of water and contaminant through the top boundary, the ground surface. The penetration depth decreases with an increasing sorption coefficient. (authors)

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Required toroidal confinement for fusion and omnigeneity

Deuterium–tritium (DT) burning requires a long energy confinement times compared to collision times, so the particle distribution functions must approximate local Maxwellians. Non-equilibrium thermodynamics is applicable, which gives relations among transport, entropy production, the collision frequency, and the deviation from a Maxwellian. The distribution functions are given by the Fokker–Planck equation, which is an advection–diffusion equation. A large hyperbolic operator, the Vlasov operator with the particle trajectories as its characteristics, equals a small diffusive operator, the collision operator. The collisionless particle trajectories would be chaotic in stellarators without careful optimization. This would lead to rapid entropy production and transport—far beyond what is consistent with a self-sustaining DT burn. Omnigeneity is the weakest general condition that is consistent with a sufficiently small entropy production associated with the thermal particle trajectories. Omnigeneity requires that the contours of constant magnetic field strength be unbounded in at least one of the two angular coordinates in magnetic surfaces and that there be a symmetry in the field-strength wells along the field lines. Even in omnigenous plasmas, fluctuations due to microturbulence can produce chaotic particle trajectories and the gyro-Bohm transport is seen in many stellarator and tokamak experiments. The higher the plasma temperature above 10 keV, the smaller the transport must be compared to gyro-Bohm for a self-sustaining DT burn. The hot alphas of DT fusion heat the electrons. When the ion–electron equilibration time is long compared to the ion energy confinement time, a self-sustaining DT burn is not possible, which sets a limit on the electron temperature.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

DPFEHM: a differentiable subsurface physics simulator

The Earth’s subsurface is a key resource that provides energy via fossil fuels and geothermal sources, stores drinking water, and is used in the fight against climate change via carbon sequestration. Simulating the physical processes that occur in the Earth’s subsurface with computers enables better use of this resource. DPFEHM is a Julia package that includes computer models with a focus on the Earth’s subsurface, especially fluid flow, which is critical for the aforementioned applications. DPFEHM is able to solve the groundwater flow equations (single phase flow), Richards equation (air/water), the advection-dispersion equation, and the 2d wave equation. One of the key features of DPFEHM is that it supports automatic differentiation, so it can be integrated into machine learning workflows using frameworks such as Flux or PyTorch. The automatic differentiation capabilities give it the same performance as adjoint methods.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Mass Transport in Membrane Systems: Flow Regime Identification by Fourier Analysis

The numerical calculation of local mass distributions in membrane systems by computational fluid dynamics (CFD) offers indispensable benefits. However, the concept to calculate such distributions in response to separate variations of operation conditions (OCs) makes it difficult to address overall, flow-physics-related questions, which require the consideration of the collective interaction of OCs. It is shown that such understanding-related relationships can be obtained by the analytical solution of the advection–diffusion equation considered. A Fourier series model (FSM) is presented, which provides exact solutions of an advection–diffusion equation for a wide range of OCs. On this basis, a new zeroth-order model is developed, which is very simple and as accurate as the complete FSM for all conditions of practical relevance. Advection-dominated blocked and diffusion-dominated unblocked flow regimes are identified (depending on a Péclet number which compares the flow geometry with a length scale imposed by the flow), which implies relevant requirements for the use of lab results for pilot- and full-scale applications. Analyses reveal the equivalence of variations of OCs, which offers a variety of options to accomplish desired flow regime changes.

Heinz, Stefan (ORCID:0000000248712416)↗
DOE OSTI2021

Toward more mechanistic representations of biogeochemical processes in stream and river networks: Implementation and demonstration of a multiscale model : Modeling Archive

This Modeling Archive supports the ORNL-SFA and IDEAS-Waterhsed publication submitted to Environmental Modeling and Software. A recently introduced multiscale model for representing the combined effect of hyporheic exchange flows and small-scale hyporheic-zone biogeochemical processes is extended from the reach scale to river network scales. The ADELS (Advection Dispersion Equation with Lagrangian Subgrid) model uses a one-dimensional advection-dispersion-reaction equation for the channel network and couples that equation at each channel location to a one-dimensional advection-reaction subgrid model representing an ensemble of streamlines that are diverted into the hyporheic zone before returning to the channel. The subgrid model is written in a computationally advantageous Lagrangian form with hyporheic age replacing the hyporheic travel distance. In the paper, we summarized implementation in the integrated surface/subsurface hydrology modeling system Advanced Terrestrial Simulator (ATS).

East Fork Poplar Creek↗
NASA NTRS1998

Algebraic Nonoverlapping Domain Decomposition Methods for Stabilized FEM and FV Discretizations

We consider preconditioning methods for convection dominated fluid flow problems based on a nonoverlapping Schur complement domain decomposition procedure for arbitrary triangulated domains. The triangulation is first partitioned into a number of subdomains and interfaces which induce a natural 2 x 2 partitioning of the p.d.e. discretization matrix. We view the Schur complement induced by this partitioning as an algebraically derived coarse space approximation. This avoids the known difficulties associated with the direct formation of an effective coarse discretization for advection dominated equations. By considering various approximations of the block factorization of the 2 x 2 system, we have developed a family of robust preconditioning techniques. A computer code based on these ideas has been developed and tested on the IBM SP2 using MPI message passing protocol. A number of 2-D CFD calculations will be presented for both scalar advection-diffusion equations and the Euler equations discretized using stabilized finite element and finite volume methods. These results show very good scalability of the preconditioner for various discretizations as the number of processors is increased while the number of degrees of freedom per processor is fixed.

Barth, Timothy J.↗
DOE OSTI2021

AIR Algebraic Multigrid for a Space-Time Hybridizable Discontinuous Galerkin Discretization of Advection(-Diffusion)

This article investigates the efficiency, robustness, and scalability of approximate ideal restriction (AIR) algebraic multigrid as a preconditioner in the all-at-once solution of a space-time hybridizable discontinuous Galerkin discretization of advection-dominated flows. The motivation for this study is that the time-dependent advection-diffusion equation can be seen as a “steady” advection-diffusion problem in $(d+1)$-dimensions and AIR has been shown to be a robust solver for steady advection-dominated problems. Numerical examples demonstrate the effectiveness of AIR as a preconditioner for advection-diffusion problems on fixed and time-dependent domains, using both slab-by-slab and all-at-once space-time discretizations, and in the context of uniform and space-time adaptive mesh refinement. A closer look at the geometric coarsening structure that arises in AIR also explains why AIR can provide robust, scalable, space-time convergence on advective and hyperbolic problems, while most multilevel parallel-in-time schemes struggle with such problems.

97 MATHEMATICS AND COMPUTING↗