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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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At least 181 records · Page 10
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations

In this work, we develop implicit rank-adaptive schemes for time-dependent matrix differential equations. The dynamic low rank approximation (DLRA) is a well-known technique to capture the dynamic low rank structure based on Dirac–Frenkel time-dependent variational principle. In recent years, it has attracted a lot of attention due to its wide applicability. Our schemes are inspired by the three-step procedure used in the rank adaptive version of the unconventional robust integrator (the so called BUG integrator) (Ceruti et al. in BIT Numer Math 62(4):1149–1174, 2022) for DLRA. First, a prediction (basis update) step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using an implicit solves for the small core matrix. Finally, a truncation is made according to a prescribed error threshold. Since the DLRA is evolving the differential equation projected on to the tangent space of the low rank manifold, the error estimate of the BUG integrator contains the tangent projection (modeling) error which cannot be easily controlled by mesh refinement. This can cause convergence issue for equations with cross terms. To address this issue, we propose a simple modification, consisting of merging the row and column spaces from the explicit step truncation method together with the BUG spaces in the prediction step. In addition, we propose an adaptive strategy where the BUG spaces are only computed if the residual for the solution obtained from the prediction space by explicit step truncation method, is too large. Here, we prove stability and estimate the local truncation error of the schemes under assumptions. We benchmark the schemes in several tests, such as anisotropic diffusion, solid body rotation and the combination of the two, to show robust convergence properties.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Parallel-in-Time Solution of Allen-Cahn Equations by Integrating Operator Learning into the Parareal Method

While recent advances in deep learning have shown promising efficiency gains in solving time-dependent partial differential equations (PDEs), matching the accuracy of conventional numerical solvers still remains a challenge. One strategy to improve the accuracy of deep learning-based solutions for time-dependent PDEs is to use the learned model as the coarse propagator in the Parareal method and a traditional numerical method as the fine solver. However, successful integration of deep learning into the Parareal method requires consistency between the coarse and fine solvers, particularly for PDEs exhibiting rapid changes such as sharp transitions. Here, to ensure this consistency, we propose using convolutional neural networks (CNNs) to learn the fully discrete time-stepping operator defined by the same numerical scheme employed as the fine solver. We demonstrate the effectiveness of the proposed method in solving the classical and mass-conservative Allen–Cahn (AC) equations. Through iterative updates in the Parareal algorithm, our approach achieves a significant computational speedup compared to traditional fine solvers while converging to high-accuracy solutions. Our results highlight that the proposed hybrid Parareal algorithm effectively accelerates simulations, particularly when implemented on multiple GPUs, and converges to the desired accuracy in only a few iterations. Another advantage of our method is that the CNN model is trained on trajectory-based data generated from random initial conditions, such that the trained model can be used to solve the AC equations with various initial conditions without retraining. This work demonstrates the potential of integrating neural network methods into parallel-in-time frameworks for efficient and accurate simulations of time-dependent PDEs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Comparison of exponential integrators and traditional time integration schemes for the shallow water equations

We report the time integration scheme is probably one of the most fundamental choices in the development of an ocean model. In this paper, we investigate several time integration schemes when applied to the shallow water equations. This set of equations is accurate enough for the modeling of a shallow ocean and is also relevant to study as it is the one solved for the barotropic (i.e. vertically averaged) component of a three dimensional ocean model. We analyze different time stepping algorithms for the linearized shallow water equations. High order explicit schemes are accurate but the time step is constrained by the Courant-Friedrichs-Lewy stability condition. Implicit schemes can be unconditionally stable but, in practice lack accuracy when used with large time steps. In this paper we propose a detailed comparison of such classical schemes with exponential integrators. The accuracy and the computational costs are analyzed in different configurations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Time-stepping DPG formulations for the heat equation

For a wide range of PDEs, the discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan provides discrete stability starting from a coarse mesh and minimization of the residual in a user-controlled norm, among other appealing features. Research on DPG for transient problems has mainly focused on spacetime discretizations, which has theoretical advantages, but practical costs for computations and software implementations. The sole examination of time-stepping DPG formulations was performed by Führer, Heuer, and Gupta, who applied Rothe’s method to an ultraweak formulation of the heat equation to develop an implicit time-stepping scheme; their work emphasized theoretical results, including error estimates in time and space. Here, we follow Führer, Heuer, and Gupta in examining the heat equation; our focus is on numerical experiments, examining the stability and accuracy of several formulations, including primal as well as ultraweak, and explicit as well as implicit and Crank–Nicolson time-stepping schemes. We are additionally interested in communication-avoiding algorithms, and we therefore include a highly experimental formulation that places all the trace terms on the right-hand side of the equation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

HomPINNs: Homotopy physics-informed neural networks for learning multiple solutions of nonlinear elliptic differential equations

Physics-informed neural networks (PINNs) based machine learning is an emerging framework for solving nonlinear differential equations. However, due to the implicit regularity of neural network structure, PINNs can only find the flattest solution in most cases by minimizing the loss functions. In this paper, we combine PINNs with the homotopy continuation method, a classical numerical method to compute isolated roots of polynomial systems, and propose a new deep learning framework, named homotopy physics-informed neural networks (HomPINNs), for solving multiple solutions of nonlinear elliptic differential equations. The implementation of an HomPINN is a homotopy process that is composed of the training of a fully connected neural network, named the starting neural network, and training processes of several PINNs with different tracking parameters. The starting neural network is to approximate a starting function constructed by the trivial solutions, while other PINNs are to minimize the loss functions defined by boundary condition and homotopy functions, varying with different tracking parameters. These training processes are regraded as different steps of a homotopy process, and a PINN is initialized by the well-trained neural network of the previous step, while the first starting neural network is initialized using the default initialization method. Finally, several numerical examples are presented to show the efficiency of our proposed HomPINNs, including reaction-diffusion equations with a heart-shaped domain.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

An implicit-in-time DPG formulation of the 1D1V Vlasov-Poisson equations

Efficient solution of the Vlasov equation, which can be up to six-dimensional, is key to the simulation of many difficult problems in plasma physics. The discontinuous Petrov-Galerkin (DPG) finite element methodology provides a framework for the development of stable (in the sense of Ladyzhenskaya–Babuška–Brezzi conditions) finite element formulations, with built-in mechanisms for adaptivity. While DPG has been studied extensively in the context of steady-state problems and to a lesser extent with space-time discretizations of transient problems, relatively little attention has been paid to time-marching approaches. In the present work, we study a first application of time-marching DPG to the Vlasov equation, using backward Euler for a Vlasov-Poisson discretization. We demonstrate adaptive mesh refinement for two problems: the two-stream instability problem, and a cold diode problem. Furthermore, we believe the present work is novel both in its application of unstructured adaptive mesh refinement (as opposed to block-structured adaptivity, which has been studied previously) in the context of Vlasov-Poisson, as well as in its application of DPG to the Vlasov-Poisson system. We also discuss extensive additions to the Camellia library in support of both the present formulation as well as extensions to higher dimensions, Maxwell equations, and space-time formulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation

Here, we study reproducing kernel (RK) collocation method for peridynamic Navier equation. In the first part, we apply a linear RK approximation to both displacement and dilatation, and then back-substitute dilatation and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit for a fixed nonlocal interaction length and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. In the second part, we apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lamé parameters satisfy λ ≥ μ to avoid extra assumptions on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.

42 ENGINEERING↗
DOE OSTI2024

Divide and conquer: Learning chaotic dynamical systems with multistep penalty neural ordinary differential equations

Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Here, our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerates the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE, is applied to chaotic systems such as the Kuramoto-Sivashinsky equation, the two-dimensional Kolmogorov flow, and ERA5 reanalysis data for the atmosphere. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.

Chaotic dynamical systems↗
DOE OSTI2025

Physics-based stabilized finite element approximations of the Poisson–Nernst–Planck equations

We present and analyze two stabilized finite element methods for solving numerically the Poisson–Nernst–Planck equations. The stabilization we consider is carried out by using a shock detector and a discrete graph Laplacian operator for the ion equations, whereas the discrete equation for the electric potential need not be stabilized. Discrete solutions stemmed from the first algorithm preserve both maximum and minimum discrete principles. For the second algorithm, its discrete solutions are conceived so that they hold discrete principles and obey an entropy law provided that an acuteness condition is imposed for meshes. Remarkably the latter is found to be unconditionally stable. We validate our methodology through transient numerical experiments that show convergence toward steady-state solutions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Learning subgrid-scale models with neural ordinary differential equations

We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary differential equations (NODEs). Solving systems with fine temporal and spatial grid scales is an ongoing computational challenge, and closure models are generally difficult to tune. Machine learning approaches have increased the accuracy and efficiency of computational fluid dynamics solvers. In this approach neural networks are used to learn the coarse- to fine-grid map, which can be viewed as subgrid-scale parameterization. We propose a strategy that uses the NODE and partial knowledge to learn the source dynamics at a continuous level. Our method inherits the advantages of NODEs and can be used to parameterize subgrid scales, approximate coupling operators, and improve the efficiency of low-order solvers. Numerical results with the two-scale Lorenz 96 ODE, the convection-diffusion PDE, and the viscous Burgers' PDE are used to illustrate this approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

TUMME: Tsinghua University Minnesota Master Equation program

We report that TUMME is a program for assembling and solving master equations for gas-phase chemical kinetics based on chemically significant eigenmodes. TUMME has interfaces to the Gaussian, Polyrate, and/or MSTor output files that allow the master equation code to obtain the microcanonical flux coefficients needed for the coefficient matrix of the master equation. The flux coefficients for reactions with barriers can be calculated by multi-structural variational transition state theory with small-curvature tunneling (MS-VTST/SCT) or by simpler approximations to this such as conventional transition state theory without tunneling (also called RRKM theory). The flux coefficients for barrierless reactions are provided by a hard-sphere model. TUMME is written in double precision with Python 3; quadruple and octuple precision are also available for some subtasks in C++. The Python code can run in serial or parallel (MP or MPI), and the C++ code can run on a single processor or on multiple processors with OpenMP.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Derivation of physical equations for high-speed laser welding using large language models

It is challenging to formulate complex physical phenomena that occur in a manufacturing process, particularly when the available data are limited, rendering conventional data-driven approaches ineffective. This study aims to predict humping onset in high-speed laser welding by introducing a novel framework, namely text-to-equations generative pre-trained transformer (T2EGPT). This method leverages the capabilities of large language models (LLMs), in combination with sparse experimental data and enriched literature data, to derive an interpretable and generalizable equation for predicting humping initiation. By capturing key correlations among physical parameters, T2EGPT generates a compact and dimensionless expression that accurately predicts hump formation. The equation reveals that humping arises from the interplay between inertia-driven backward melt flow and capillary-driven surface stabilization, where inertial forces drive molten metal backward and capillary forces resist surface deformation. Furthermore, compared to traditional data-driven models, T2EGPT demonstrates enhanced predictive accuracy and cross-material transferability. More broadly, this study highlights the potential of LLMs to integrate textual information with data-driven discovery, enabling the extraction of physical laws in data-scarce scientific domains.

36 MATERIALS SCIENCE↗
DOE OSTI2023

A sharp immersed method for 2D flow-body interactions using the vorticity-velocity Navier-Stokes equations

Immersed methods discretize boundary conditions for complex geometries on background Cartesian grids. Here, this makes such methods especially suitable for two-way coupled flow-body problems, where the body mechanics are partially driven by hydrodynamic forces. However, for the vorticity-velocity form of the Navier-Stokes equations, existing immersed geometry discretizations for two-way coupled problems only achieve first order spatial accuracy near solid boundaries. Here we introduce a sharp-interface approach based on the immersed interface method to handle the one- and two-way coupling between an incompressible flow and one or more rigid bodies using the 2D vorticity-velocity Navier-Stokes equations. Our main contributions are three-fold. First, we develop and analyze a moving boundary treatment for sharp immersed methods that can be applied to PDEs with implicitly defined boundary conditions, such as those commonly imposed on the vorticity field. Second, we develop a two-way coupling methodology for the vorticity-velocity Navier-Stokes equations based on control-volume momentum balance that does not require the pressure field. Third, we show through extensive testing and validation that our resulting flow-body solver reaches second-order accuracy for most practical scenarios, and provides significant efficiency benefits compared to a representative first-order approach.

42 ENGINEERING↗
DOE OSTI2024

Anti-symmetric and positivity preserving formulation of a spectral method for Vlasov-Poisson equations

We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Reduced order models for thermal radiative transfer problems based on moment equations and data-driven approximations of the Eddington tensor

Here a new group of structure and asymptotic preserving reduced-order models (ROMs) for multidimensional nonlinear thermal radiative transfer (TRT) problems is presented. They are formulated by means of the nonlinear projective approach and data compression techniques. The nonlinear projection is applied to the Boltzmann transport equation (BTE) to derive a hierarchy of low-order moment equations. Approximation of the Eddington tensor that provides exact closure for the system of moment equations is found with projection-based data-driven methodologies. These include the (i) proper orthogonal decomposition (POD), (ii) dynamic mode decomposition (DMD) and (iii) a variant of the DMD. A parameterization is derived for this ROM for the temperature of radiation incoming to the problem domain (the radiation drive temperature). This parameterization is informed from results of a dimensionless study of the TRT problem. Analysis of the ROMs is performed on the classical Fleck-Cummings TRT multigroup test problem in 2D geometry with a radiation-driven Marshak wave. Numerical results are presented to demonstrate the performance of these ROMs for the simulation of evolving radiation and heat waves. Results show these models to be sufficiently accurate for practical computations with rather low-rank representations of the Eddington tensor. As the rank of the approximation is increased, the errors of solutions generated by the ROMs gradually decreases.

42 ENGINEERING↗
DOE OSTI2024

Collocation methods for nonlinear differential equations on low-rank manifolds

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A Poisson equation method for prescribing fully developed non-Newtonian inlet conditions for computational fluid dynamics simulations in models of arbitrary cross-section

Prescribing inlet boundary conditions for computational fluid dynamics (CFD) simulations of internal flow in complex geometries such as anatomical vascular models is challenging. In the absence of patient-specific inlet velocity data, a common approach for long blood vessels is to assume that the inlet flow is fully developed. In vessels of irregular cross section, however, prescribing fully developed conditions is complicated due to the lack of a general closed-form analytical solution. In this study, we develop a simple Poisson equation method for prescribing fully developed inlet conditions for the flow of either Newtonian or non-Newtonian fluids in CFD models of arbitrary cross-section. We first derive the generalized Poisson equation for fully developed flow of a non-Newtonian fluid and we then develop and verify a methodology for numerically computing the solution on any planar boundary domain. In addition, we develop a simple extension of the method for prescribing a non-orthogonal inlet velocity that represents fully developed flow from an upstream tube that is connected to the CFD inlet at a non-orthogonal angle. This may be used to investigate a common source of uncertainty in CFD simulations of internal flow that is due to a lack of information concerning the exact streamwise flow direction at the inlets. Comparison to several Newtonian and non-Newtonian benchmark verification solutions shows the method to be extremely accurate. As a practical demonstration case, we use the method to prescribe fully developed conditions on multiple non-circular inlets for the non-Newtonian flow of blood in a patient-specific model of the inferior vena cava (IVC). Finally, we further demonstrate the utility of the method by performing a sensitivity study using the patient-specific IVC model, wherein we investigate the influence of inlet velocity flow direction on the non-Newtonian IVC hemodynamics. Given its simplicity and computational efficiency, the method is shown to be far superior to alternative approaches for prescribing fully developed inlet conditions in such complicated geometries. In conclusion, to facilitate the adoption of our Poisson equation method, we have distributed our OpenFOAM source code and the associated test cases from this study as open-source software.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

An ideal MHD δW stability analysis that bypasses the Newcomb equation

In the following work, we demonstrate the efficacy of a Riccati δW ideal MHD stability analysis that bypasses the numerically intractable integration of the Newcomb equation. By transforming the linear Newcomb equation into a quadratic Riccati equation, an accurate and equivalent δW analysis is performed that is shown to enjoy some numerical advantages. Additionally, we demonstrate that the Riccati approach is better conditioned than its Newcomb counterpart at the magnetic axis, and we apply dynamical systems insights to examine its behavior at singular points. We further discuss the constraints involved in solving for the extremal admissible perturbations of a δW analysis.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗