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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 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
NASA NTRS1975

The geometric approach to sets of ordinary differential equations and Hamiltonian dynamics

The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field V in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along V: scalar fields, forms, vectors, and integrals over subspaces. It is shown that to any field V can be associated a Hamiltonian structure of forms if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added. Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton's variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms must include the characteristics of the set.

Estabrook, F. B.↗
NASA NTRS1980

Nonlinear grid error effects on numerical solution of partial differential equations

Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.

Dey, S. K.↗
NASA NTRS1982

Marching grid generation using parabolic partial differential equations

The feasibility of using parabolic partial differential equations for grid generation is examined. Source terms in the form of a linear interpolation between the current grid and the outer boundary are used in generating a grid for a two-dimensional airfoil flow. Grid generation equations are derived by assuming that grid spacings are locally nonuniform on the computational domain. The grid spacing control method is described in detail, and the local orthogonality of the grid lines is discussed. An O-mesh and an H-mesh generated by the described method are shown.

Nakamura, S.↗
NASA NTRS1972

Some efficient methods for obtaining infinite series solutions of n-th order linear ordinary differential equations

The use of the theta-operator method and generalized hypergeometric functions in obtaining solutions to nth-order linear ordinary differential equations is explained. For completeness, the analysis of the differential equation to determine whether the point of expansion is an ordinary point or a regular singular point is included. The superiority of the two methods shown over the standard method is demonstrated by using all three of the methods to work out several examples. Also included is a compendium of formulae and properties of the theta operator and generalized hypergeometric functions which is complete enough to make the report self-contained.

Allen, G.↗
NASA NTRS1978

On the accuracy and convergence of implicit numerical integration of finite element generated ordinary differential equations

A study of accuracy and convergence of linear functional finite element solution to linear parabolic and hyperbolic partial differential equations is presented. A variable-implicit integration procedure is employed for the resultant system of ordinary differential equations. Accuracy and convergence is compared for the consistent and two lumped assembly procedures for the identified initial-value matrix structure. Truncation error estimation is accomplished using Richardson extrapolation.

Baker, A. J.↗
DOE OSTI2023

Loop-by-loop differential equations for dual (elliptic) Feynman integrals

We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an ε-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of q-series. Then, we discuss interesting properties satisfied by our dual basis as well as its simple relation to the known ε-form basis of Feynman integrands. The underlying K3-geometry of the three-loop four-mass sunrise integral is also discussed. Finally, we speculate on how to construct a “good” loop-by-loop basis at three-loop.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS1972

Numerical method for the solution of large systems of differential equations of the boundary layer type

A numerical method for the solution of large systems of nonlinear differential equations of the boundary-layer type is described. The method is a modification of the technique for satisfying asymptotic boundary conditions. The present method employs inverse interpolation instead of the Newton method to adjust the initial conditions of the related initial-value problem. This eliminates the so-called perturbation equations. The elimination of the perturbation equations not only reduces the user's preliminary work in the application of the method, but also reduces the number of time-consuming initial-value problems to be numerically solved at each iteration. For further ease of application, the solution of the overdetermined system for the unknown initial conditions is obtained automatically by applying Golub's linear least-squares algorithm. The relative ease of application of the proposed numerical method increases directly as the order of the differential-equation system increases. Hence, the method is especially attractive for the solution of large-order systems. After the method is described, it is applied to a fifth-order problem from boundary-layer theory.

Green, M. J.↗
NASA NTRS1994

Algorithms For Integrating Nonlinear Differential Equations

Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.

Freed, A. D.↗
DOE OSTI2023

Improved quantum algorithms for linear and nonlinear differential equations

We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes the run time of quantum algorithms for linear ODEs opening the door to an application to a wider class of linear and nonlinear ODEs. In [1], a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices including singular matrices. The algorithm here is also exponentially faster than the bounds derived in [1] for certain classes of diagonalizable matrices. Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in [2]). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by [3], but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas [2] and [3] additionally require normality.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗