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NASA NTRS1987

On Lipschitz continuity of nonlinear differential operators

In connection with approximations for nonlinear evolution equations, it is standard to assume that nonlinear terms are at least locally Lipschitz continuous. However, it is shown here that f = f(X,del sub u(X)) is Lipschitz continuous from the subspace W sup 1, infinity is a subset of L sub 2 into W sup 1,2, and maps W sup 2, infinity into W sup 1, infinity, if and only if f is affine with W sup 1, infinity coefficients. In fact, a local version of this claim is proved.

Keeling, Stephen L.↗
NASA NTRS1991

The symbolic computation and automatic analysis of trajectories

Research was generally done on computation of trajectories of dynamical systems, especially control systems. Algorithms were further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear control systems. An initial design was completed of the system architecture for software to analyze nonlinear control systems using data base computing.

Grossman, Robert↗
NASA NTRS1995

Low-frequency variations in the atmospheric branch of the global hydrological cycle

The annual variation of the hydrological cycle is illustrated in terms of hemispheric-mean hydrological variables for the Northern and Southern Hemispheres, while the intraseasonal variations of the global hydrological cycle are illustrated with mean values over two hemispheres that form an east-west partition of the globe. This partition is defined by the 60 deg E-120 deg W great circle and was chosen so that the mean precipitation difference and the divergent water vapor transport between the two hemispheres was maximized. Two years (1979-80) of daily precipitation estimates from the Goddard Laboratory for Atmospheres an 14 years (1979-92) of upper-air data generated by the Global Data Assimilation System at the National Meteorological Center are used in making quantitative estimates of the annual and intraseasonal variations in the global hydrological cycle. The annual variations in hemispheric-mean precipitation (P-circumflex) and water vapor flux divergence (del(vector differential operator) dot Q-circumflex) for the Northern and Southern Hemispheres are comparable with amplitudes of about 0.5 approximately 0.7 mm/day. Both (P-circumflex) and (del(vector differential operator) dot Q-circumflex) vary annually in a coherent way in each hemisphere so that water vapor diverges from the winter hemisphere, where (P-circumflex) reaches its minimum, to the summer hemisphere, where (P-circumflex) attains its maximum. In fact, the hemispheric-mean divergence of water vapor flux changes sign during the annual cycle. Intraseasonal variations of hemispheric-mean precipitation mean P-tilde, evaporation mean E-tilde, and water vapor flux divergence mean del (vector differential operator) dot Q-tilde in the two hemisphres in the east-west direction are comparable with amplitudes of about 0.1 approximately 0.2 mm/day, although amplitudes in some cases exceed 0.3 mm/day. Hemispheric-mean precipitation mean P-tilde varies coherently in opposite phase for the two hemispheres, while mean del (vector differential operator) dot Q-tilde varies so that water vapor diverges from the hemisphere of maximum mean P-tilde to the hemisphere to the hemisphere of minimum mean P-tilde. Intraseasonal variations of mean P-tilde, mean E-tilde, and mean del (vector differential oprator) dot Q-tilde are in accord with the eastward propagation of the intraseasonal global divergent circulation.

Chen, Tsing-Chang↗
NASA NTRS1980

Newton's method: A link between continuous and discrete solutions of nonlinear problems

Newton's method for nonlinear mechanics problems replaces the governing nonlinear equations by an iterative sequence of linear equations. When the linear equations are linear differential equations, the equations are usually solved by numerical methods. The iterative sequence in Newton's method can exhibit poor convergence properties when the nonlinear problem has multiple solutions for a fixed set of parameters, unless the iterative sequences are aimed at solving for each solution separately. The theory of the linear differential operators is often a better guide for solution strategies in applying Newton's method than the theory of linear algebra associated with the numerical analogs of the differential operators. In fact, the theory for the differential operators can suggest the choice of numerical linear operators. In this paper the method of variation of parameters from the theory of linear ordinary differential equations is examined in detail in the context of Newton's method to demonstrate how it might be used as a guide for numerical solutions.

Thurston, G. A.↗
NASA NTRS1989

Labeled trees and the efficient computation of derivations

The effective parallel symbolic computation of operators under composition is discussed. Examples include differential operators under composition and vector fields under the Lie bracket. Data structures consisting of formal linear combinations of rooted labeled trees are discussed. A multiplication on rooted labeled trees is defined, thereby making the set of these data structures into an associative algebra. An algebra homomorphism is defined from the original algebra of operators into this algebra of trees. An algebra homomorphism from the algebra of trees into the algebra of differential operators is then described. The cancellation which occurs when noncommuting operators are expressed in terms of commuting ones occurs naturally when the operators are represented using this data structure. This leads to an algorithm which, for operators which are derivations, speeds up the computation exponentially in the degree of the operator. It is shown that the algebra of trees leads naturally to a parallel version of the algorithm.

Grossman, Robert↗
NASA NTRS1994

An Efficient Spectral Method for Ordinary Differential Equations with Rational Function Coefficients

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple three-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e. matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation N, is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.

Coutsias, Evangelos A.↗
NASA NTRS1992

The explicit computation of integration algorithms and first integrals for ordinary differential equations with polynomials coefficients using trees

This note is concerned with the explicit symbolic computation of expressions involving differential operators and their actions on functions. The derivation of specialized numerical algorithms, the explicit symbolic computation of integrals of motion, and the explicit computation of normal forms for nonlinear systems all require such computations. More precisely, if R = k(x(sub 1),...,x(sub N)), where k = R or C, F denotes a differential operator with coefficients from R, and g member of R, we describe data structures and algorithms for efficiently computing g. The basic idea is to impose a multiplicative structure on the vector space with basis the set of finite rooted trees and whose nodes are labeled with the coefficients of the differential operators. Cancellations of two trees with r + 1 nodes translates into cancellation of O(N(exp r)) expressions involving the coefficient functions and their derivatives.

Crouch, P. E.↗
NASA NTRS1973

PLZT block data composers operated in differential phase mode

The use of PLZT ceramics with the 7/65/35 composition in block data composer (BDC) input devices for holographic memory systems has previously been described for operation in the strain biased, scattering, and edge effect modes. A new and promising mode of BDC operation is the differential phase mode in which each element of a matrix array BDC acts as a phase modulator. The phase modulation results from a phase difference in the optical path length between the electrically poled and depoled states of the PLZT. It is shown that a PLZT BDC can be used as a matrix-type phase modulator to record and process digital data by the differential phase mode in a holographic recording/processing system with readout contrast ratios of between 10:1 and 15:1. The differential phase mode has the advantages that strain bias is not required and that the thickness and strain variations in the PLZT are cancelled out.

Drake, M. D.↗
NASA NTRS1975

Generalized Du Fort-Frankel methods for parabolic initial boundary value problems

The Du Fort-Frankel difference scheme is generalized to difference operators of arbitrary high order accuracy in space and to arbitrary order of the parabolic differential operator. Spectral methods can also be used to approximate the spatial part of the differential operator. The scheme is explicit, and it is unconditionally stable for the initial value problem. Stable boundary conditions are given for two different fourth order accurate space approximations.

Gottlieb, D.↗
NASA NTRS1976

Generalized Du Fort-Frankel methods for parabolic initial-boundary value problems

The Du Fort-Frankel difference scheme is generalized to difference operators of arbitrary high order accuracy in space and to arbitrary order of the parabolic differential operator. Spectral methods can also be used to approximate the spatial part of the differential operator. The scheme is explicit, and it is unconditionally stable for the initial value problem. Stable boundary conditions are given for two different fourth order accurate space approximations.

Gottlieb, D.↗
NASA NTRS1989

Flight-test evaluation of civil helicopter terminal approach operations using differential GPS

A civil code differential Global Positioning System (DGPS) has been developed and flight-tested by the NASA Ames Research Center. The system was used to evaluate the performance of the DGPS for support of helicopter terminal approach operations. The airborne component of the DGPS was installed in a NASA helicopter. The ground-reference component was installed in a mobile van and equipped with a real-time VHF telemetry data link to transmit correction information to the aircraft system. An extensive series of tests was conducted to evaluate the performance of the system for several different configurations of the airborne navigation filter. This paper will describe the systems, the results of the flight tests, and the results of the posttest analysis.

Edwards, F. G.↗
NASA NTRS1985

Civil helicopter flight operations using differential GPS

The results of NASA flight trials of a dual-receiver differential global positioning system (DGPS) for civilian helicopter navigation applications, are presented. The three principal components of the DGPS system are described, including the GPS ground-reference system, a range tracking system, and an on-board sequential GPS receiver. The ground-based receiver in the DGPS operates at a known fixed location and receives C/A code signals from NAVSTAR satellites. System bias errors in the ground receiver are subtracted from the airborne solution for the navigational fix. Calculations of the differential bias error are carried out using an on-board PDP-11/34 M research computer. The ground-reference differential corrections for satellites are given in a table. It is shown that the differential correction signal of the DGPS contains only a small (0.1 rad/sec) high-frequency component which can be attributed to system error. A schematic diagram of the DGPS postflight data processing routine is provided.

Edwards, F. G.↗
NASA NTRS1989

Modified Chebyshev pseudospectral method with O(N exp -1) time step restriction

The extreme eigenvalues of the Chebyshev pseudospectral differentiation operator are O(N exp 2) where N is the number of grid points. As a result of this, the allowable time step in an explicit time marching algorithm is O(N exp -2) which, in many cases, is much below the time step dictated by the physics of the partial differential equation. A new set of interpolating points is introduced such that the eigenvalues of the differentiation operator are O(N) and the allowable time step is O(N exp -1). The properties of the new algorithm are similar to those of the Fourier method. The new algorithm also provides a highly accurate solution for non-periodic boundary value problems.

Kosloff, Dan↗