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At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Solving high-dimensional partial integral differential equations: The finite expression method

Partial integro-differential equations (PIDEs) have broad applications in the sciences, from electro-magnetism to options pricing. Here, in this paper, we introduce a new finite expression method (FEX) to solve PIDEs. This approach builds upon the original FEX and its inherent advantages with new advances: 1) A novel method of parameter grouping is proposed to reduce the number of coefficients in high-dimensional function approximation; 2) A Taylor series approximation method is implemented to significantly improve the computational efficiency and accuracy of the evaluation of the integral terms of PIDEs. The new FEX based method, denoted FEX-PG to indicate the addition of the parameter grouping (PG) step to the algorithm, provides both high accuracy and interpretable numerical solutions, with the outcome being an explicit equation that facilitates intuitive understanding of the underlying solution structures. These features are often absent in traditional methods, such as finite element methods (FEM) and finite difference methods, as well as in deep learning-based approaches. To benchmark our method against recent advances, we apply the new FEX-PG to solve benchmark PIDEs in the literature. In high-dimensional settings, FEX-PG exhibits strong and robust performance, achieving relative errors on the order of single precision machine epsilon, significantly outperforming existing approaches based on neural networks.

Combinatorial optimization↗
NASA NTRS1985

Topics in spectral methods

After detailing the construction of spectral approximations to time-dependent mixed initial boundary value problems, a study is conducted of differential equations of the form 'partial derivative of u/partial derivative of t = Lu + f', where for each t, u(t) belongs to a Hilbert space such that u satisfies homogeneous boundary conditions. For the sake of simplicity, it is assumed that L is an unbounded, time-independent linear operator. Attention is given to Fourier methods of both Galerkin and pseudospectral method types, the Galerkin method, the pseudospectral Chebyshev and Legendre methods, the error equation, hyperbolic partial differentiation equations, and time discretization and iterative methods.

Gottlieb, D.↗
NASA NTRS1974

The modified equation approach to the stability and accuracy analysis of finite-difference methods

The stability and accuracy of finite-difference approximations to simple linear partial differential equations are analyzed by studying the modified partial differential equation. Aside from round-off error, the modified equation represents the actual partial differential equation solved when a numerical solution is computed using a finite-difference equation. The modified equation is derived by first expanding each term of a difference scheme in a Taylor series and then eliminating time derivatives higher than first order by certain algebraic manipulations. The connection between 'heuristic' stability theory based on the modified equation approach and the von Neumann (Fourier) method is established. In addition to the determination of necessary and sufficient conditions for computational stability, a truncated version of the modified equation can be used to gain insight into the nature of both dissipative and dispersive errors.

Warming, R. F.↗
DOE OSTI2024

Accurate data-driven surrogates of dynamical systems for forward propagation of uncertainty

Stochastic collocation (SC) is a well-known non-intrusive method of constructing surrogate models for uncertainty quantification. In dynamical systems, SC is especially suited for full-field uncertainty propagation that characterizes the distributions of the high-dimensional solution fields of a model with stochastic input parameters. However, due to the highly nonlinear nature of the parameter-to-solution map in even the simplest dynamical systems, the constructed SC surrogates are often inaccurate. Here, this work presents an alternative approach, where we apply the SC approximation over the dynamics of the model, rather than the solution. By combining the data-driven sparse identification of nonlinear dynamics framework with SC, we construct dynamics surrogates and integrate them through time to construct the surrogate solutions. We demonstrate that the SC-over-dynamics framework leads to smaller errors, both in terms of the approximated system trajectories as well as the model state distributions, when compared against full-field SC applied to the solutions directly. We present numerical evidence of this improvement using three test problems: a chaotic ordinary differential equation, and two partial differential equations from solid mechanics.

42 ENGINEERING↗
NASA NTRS1975

Relaxation methods in fluid mechanics

The present work considers the iterative solution of a coupled set of difference equations and examines methods that carry successive approximates to a state that is invariant with further iteration and independent of the initial guess. Methods are studied with regard to their efficiency and economy of computer resources. The basic principles of classical relaxation are set forth, with attention confined to linear elliptic equations. This discussion involves the evaluation of the spectral radius that is the magnitude of the eigenvalue with largest modulus. The subject of relaxation is then related to the study of ordinary differential equations and hyperbolic partial differential equations. Problems that occur when linearly dependent eigenvectors appear in the relaxation matrix are discussed, leading to multiply connected eigenvalues in the Jordan canonical form. Finally, a brief survey of relaxation methods used in aerodynamics is given.

Lomax, H.↗
NASA NTRS1995

On the Dynamics of Some Discretizations of Convection-Diffusion Equations

Numerical discretizations of differential equations which model physical processes can possess dynamics quite different from that of the equations themselves. Recently the emphasis has been on the the dynamics of numerical discretizations for Ordinary Differential Equations (ODEs). For Partial Differential Equations (PDEs) using a method of lines approach the situation is more complex. First, the spatial discretisation may introduce dynamics not present in the original equations; second, the solution of the resulting system of ODEs is open to the modified dynamics of the ODE solver used. These two effects may interact in a complex manner. In this talk we present some results of our recent work on the dynamics of discretizations of convection-diffusion equations, including those produced using Total Variation Diminishing (TVD) schemes and adaptive grid techniques. A more general overview of the area may be found on our accompanying poster presentation.

Sweby, Peter K.↗
NASA NTRS1978

An extension of A-stability to alternating direction implicit methods

An alternating direction implicit (ADI) scheme was constructed by the method of approximate factorization. An A-stable linear multistep method (LMM) was used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation. Sufficient conditions for the A-stability of the LMM were determined by applying the theory of positive real functions to reduce the stability analysis of the partial differential equations to a simple algebraic test. A linear test equation for partial differential equations is defined and then used to analyze the stability of approximate factorization schemes. An ADI method for the three-dimensional heat equation is also presented.

Warming, R. F.↗
DOE OSTI2024

BPZ equations for higher degenerate fields and nonperturbative Dyson-Schwinger equations

In the two-dimensional Liouville conformal field theory, correlation functions involving a degenerate field satisfy partial differential equations due to the decoupling of the null descendant field. On the other hand, the instanton partition function of a four-dimensional N = 2 supersymmetric theory in the Ω -background at a special point of the parameter space also satisfies a partial differential equation resulting from the constraints of the gauge field configurations. This partial differential equation can be proved using the nonperturbative Dyson-Schwinger equations. We show for the next-to-simplest case that the partial differential equations obtained from two different perspectives can be identified, thereby confirming an assertion of the Bogomol’nyi-Prasad-Sommerfield/conformal field theory correspondence. Published by the American Physical Society 2024

Astronomy & Astrophysics↗
NASA NTRS1998

Self-Similar Compressible Free Vortices

Lie group methods are used to find both exact and numerical similarity solutions for compressible perturbations to all incompressible, two-dimensional, axisymmetric vortex reference flow. The reference flow vorticity satisfies an eigenvalue problem for which the solutions are a set of two-dimensional, self-similar, incompressible vortices. These solutions are augmented by deriving a conserved quantity for each eigenvalue, and identifying a Lie group which leaves the reference flow equations invariant. The partial differential equations governing the compressible perturbations to these reference flows are also invariant under the action of the same group. The similarity variables found with this group are used to determine the decay rates of the velocities and thermodynamic variables in the self-similar flows, and to reduce the governing partial differential equations to a set of ordinary differential equations. The ODE's are solved analytically and numerically for a Taylor vortex reference flow, and numerically for an Oseen vortex reference flow. The solutions are used to examine the dependencies of the temperature, density, entropy, dissipation and radial velocity on the Prandtl number. Also, experimental data on compressible free vortex flow are compared to the analytical results, the evolution of vortices from initial states which are not self-similar is discussed, and the energy transfer in a slightly-compressible vortex is considered.

vonEllenrieder, Karl↗
NASA NTRS1972

Dissipative systems.

A theory is presented that shows how the concept of dissipative systems of ordinary differential equations can be extended to include a broad class of functional and partial differential equations, such as retarded functional differential equations and parabolic partial differential equations. Since the basic hypotheses are all in terms of boundedness, finding sufficient conditions in terms of Liapunov functions would not be too difficult.

Lasalle, J. P.↗
NASA NTRS1981

Determination of the potential in a synodic system

Determination of the potential field in a fixed (inertial) system may be accomplished by the solution of a homogeneous linear partial differential equation when a family of orbits of a body moving in the field is given. This partial differential equation was presented and thoroughly analyzed earlier. The present paper discusses the same problem in a rotating system where the centrifugal and Coriolis effects render the pertinent partial differential equation in general non-homogeneous and non-linear. A linear, though non-homogeneous, partial differential equation for the determination of the synodic potential is obtained only in the special case of iso-energetic families of orbits.

Szebehely, V.↗
DOE OSTI2024

Computationally efficient and error aware surrogate construction for numerical solutions of subsurface flow through porous media

Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution. Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O ( n log n ) instead of the typical O ( n 3 ) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy’s equation and the numerical solution. Finally, although our method is derived for building a surrogate for the solution of Darcy’s equation with a random permeability field, the framework broadly applies to solutions of other partial differential equations with random coefficients.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

SUNDIALS time integrators for exascale applications with many independent systems of ordinary differential equations

Many complex systems can be accurately modeled as a set of coupled time-dependent partial differential equations (PDEs). However, solving such equations can be prohibitively expensive, easily taxing the world’s largest supercomputers. One pragmatic strategy for attacking such problems is to split the PDEs into components that can more easily be solved in isolation. This operator splitting approach is used ubiquitously across scientific domains, and in many cases leads to a set of ordinary differential equations (ODEs) that need to be solved as part of a larger “outer-loop” time-stepping approach. The SUNDIALS library provides a plethora of robust time integration algorithms for solving ODEs, and the U.S. Department of Energy Exascale Computing Project (ECP) has supported its extension to applications on exascale-capable computing hardware. In this paper, we highlight some SUNDIALS capabilities and its deployment in combustion and cosmology application codes (Pele and Nyx, respectively) where operator splitting gives rise to numerous, small ODE systems that must be solved concurrently.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1976

Computation of turbulent flows

The essential ingredients and the effectiveness of several levels of turbulent-flow partial differential equation models are considered. Zero-equation models use only the partial differential equation (pde) for the mean velocity field and do not employ turbulence pde's. One-equation models make use of an additional pde relating to the turbulence velocity scale. Attention is also given to two-equation models, stress-equation models, and large eddy simulations. Large-eddy simulations are concerned with a three-dimensional time-dependent numerical computation of the large-scale turbulence.

Reynolds, W. C.↗
NASA NTRS1984

Mathematical analysis

Simple continuum models used in the design, analysis, and control of large space structures are examined. Particular emphasis is placed on boundary value problems associated with the Load Correction Method and control problems involving partial differential equations for the large space structure models. Partial differential equations will be used to model a large space structure, base the design of an optimal controller on this model, approximate the resulting optimal control model, and compare the results with data from other methods.

Donaldson, J. A.↗
NASA NTRS1971

Similarity analysis of differential equations by Lie group.

Methods for transforming partial differential equations into forms more suitable for analysis and solution are investigated. The idea of Lie's infinitesimal contact transformation group is introduced to develop a systematic method which involves mostly algebraic manipulations. A thorough presentation of the application of this general method to the problem of similarity analysis in a broader sense - namely, the similarity between partial and ordinary differential equations, boundary value and initial value problems, and nonlinear and linear equations - is given with new and very general methods evolved for deriving the possible groups of transformations.

Na, T. Y.↗