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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
NASA NTRS1973

On the stability of numerical integration routines for ordinary differential equations.

Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.

Glover, K.↗
NASA NTRS1950

Flutter of a uniform wing with an arbitrarily placed mass according to a differential-equation analysis and a comparison with experiment

A method is presented for the calculation of the flutter speed of a uniform wing carrying an arbitrarily placed concentrated mass. The method, an extension of recently published work by Goland and Luke, involves the solution of the differential equations of motion of the wing at flutter speed and therefore does not require the assumption of specific normal modes of vibration. The order of the flutter determinant to be solved by this method depends upon the order of the system of differential equations and not upon the number of modes of vibration involved. The differential equations are solved by operational methods, and a brief discussion of operational methods as applied to boundary-value problems is included in one of two appendixes. A comparison is made with experiment for a wing with a large eccentrically mounted weight and good agreement is obtained. Sample calculations are presented to illustrate the method; and curves of amplitudes of displacement, torque, and shear for a particular case are compared with corresponding curves computed from the first uncoupled normal modes.

Runyan, Harry L↗
NASA NTRS1974

Changing stepsize in the integration of differential equations using modified divided differences

Multistep methods for solving differential equations based on numerical integration formulas or numerical differentiation formulas (for stiff equations) require special provision for changing the stepsize. New algorithms are given which make the use of modified divided differences an attractive way to carry out the change in stepsize for such methods. Error estimation and some of the important factors in stepsize selection and the selection of integration order are also considered.

Krogh, F. T.↗
NASA NTRS1975

NASTRAN solutions of problems described by simultaneous parabolic differential equations

NASTRAN solution techniques are shown for a numerical analysis of a class of coupled vector flow processes described by simultaneous parabolic differential equations. To define one physical problem type where equations of this form arise, the differential equations describing the coupled transfers of heat and mass in mechanical equilibrium with negligible mass average velocity are presented and discussed. Also shown are the equations describing seepage when both electrokinetic and hydrodynamic forces occur. Based on a variational statement of the general problem type, the concepts of scalar transfer elements and parallel element systems are introduced. It is shown that adoptation of these concepts allows the direct use of NASTRAN's existing Laplace type elements for uncoupled flow (the heat transfer elements) for treating multicomponent coupled transfer. Sample problems are included which demonstrate the application of these techniques for both steady-state and transient problems.

Mason, J. B.↗
NASA NTRS1992

Optimal moving grids for time-dependent partial differential equations

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of PDE solutions in the least-squares norm are reported.

Wathen, A. J.↗
NASA NTRS2002

Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

In this paper we review the existing and develop new continuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L(exp 2) stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.

Yan, Jue↗
DOE OSTI2026

A score-based diffusion model approach for adaptive learning of stochastic partial differential equation solutions

In this paper, we propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central role in modeling complex physical systems under uncertainty, but their numerical solutions often suffer from model errors and reduced accuracy due to incomplete physical knowledge and environmental variability. To address these challenges, we encode the governing physics into the score function of a diffusion model using simulation data and incorporate observational information via a likelihood-based correction in a reverse-time stochastic differential equation. This enables adaptive learning through iterative refinement of the solution as new data becomes available. To improve computational efficiency in high-dimensional settings, we introduce the ensemble score filter, a training-free approximation of the score function designed for real-time inference. Numerical experiments on benchmark SPDEs demonstrate the accuracy and robustness of the proposed method under sparse and noisy observations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equations

Physics-informed neural networks have emerged as an alternative method for solving partial differential equations. However, for complex problems, the training of such networks can still require high-fidelity data which can be expensive to generate. To reduce or even eliminate the dependency on high-fidelity data, we propose a novel multi-fidelity architecture which is based on a feature space shared by the low- and high-fidelity solutions. In the feature space, the projections of the low-fidelity and high-fidelity solutions are adjacent by constraining their relative distance. The feature space is represented with an encoder and its mapping to the original solution space is effected through a decoder. In conclusion, the proposed multi-fidelity approach is validated on forward and inverse problems for steady and unsteady problems described by partial differential equations.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1997

Runge-Kutta Methods for Linear Ordinary Differential Equations

Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODES) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of the Runge-Kutta method must satisfy. This freedom is used to develop methods which are more efficient than conventional Runge-Kutta methods. A fourth-order method is presented which uses only two memory locations per dependent variable, while the classical fourth-order Runge-Kutta method uses three. This method is an excellent choice for simulations of linear wave phenomena if memory is a primary concern. In addition, fifth- and sixth-order methods are presented which require five and six stages, respectively, one fewer than their conventional counterparts, and are therefore more efficient. These methods are an excellent option for use with high-order spatial discretizations.

Zingg, David W.↗