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NASA NTRS1985

Stability of finite-difference models containing two boundaries or interfaces

The stability of finite difference models of hyperbolic initial boundary value problems is connected with the propagation and reflection of parasitic waves. Wave propagation ideas are applied to models containing two boundaries or interfaces, where repeated reflection of trapped wave packets is a potential new source of instability. Various known instability phenomena are accounted for in a unified way. Results show: (1) dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface; (2) algebraic GKS instabilities can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present; and (3) GKS-stability and P-stability can be established in certain problems by showing that all numerical reflection coefficients have modulus less than 1.

Trefethen, L. N.↗

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 NTRS1986

On a large time-step high resolution scheme

This paper presents a class of new second-order accurate (2K + 3)-point explicit schemes for the computation of weak solutions of hyperbolic conservation laws, that are total-variation-diminishing under a Courant-Friedrichs-Lewy restriction of K. These highly nonlinear schemes are obtained by applying a nonoscillatory first-order accurate (2K + 1)-point scheme to a modified flux. The derived second-order accurate schemes achieve high resolutions, while retaining the robustness of the original first-order accurate scheme.

Harten, A.↗

NASA NTRS1986

Technology and the Hubble Space Telescope

The components of the Hubble Space Telescope which is a 13.1 m long, 4.27 m in diameter, and weighs 11,000 kg are described. The telescope will be placed in a 600-km circular orbit of 28.5 deg inclination and it has an expected service life of 15 years. The optical telescope assembly contains the hyperbolic Ritchey-Chretin type primary and secondary mirrors and the optical control system. The telescope is to have a field of view of 28 arcmin in diameter. The use of the fine guidance sensors for pointing stability is examined. The optical and scientific system of the telescope are contained in the support system module. A wide-field/planetary camera, ESA-furnished faint object camera, the high-speed photometer, faint-object spectrograph, high resolution spectrograph, and ESA-furnished solar arrays are located on the telescope. The maintenance and refurbishment of the telescope are discussed. The precise lambda/20 optics, stringent cleanliness, and stable pointing of the telescope produce good image quality for astronomical and solar system science studies.

Mitchell, R. E.↗

NASA NTRS1985

Finite element algorithms for compressible flow computation on a supercomputer

Methods of applying computational fluid dynamics techniques to engineering problems are discussed. The linking factors between theoretical and applied research are the shape and weight functions which produce the Galerkin finite element schemes. The flow is unsteady, viscous, heat conducting and compressible, and steady-state flows are the asymptotic limit of unsteady flows. All flows are described by the time-averaged Navier-Stokes equations (NSE) with mass and energy conservation. Local curvilinear intrinsic coordinates are applied to discretize the NSE in arbitrary geometric domains. A modified weighted residuals approach defines the discrete analogs of the physical systems modeled and a general interpolants method is used to derive families of numerical models, both implicit and explicit, finite difference and finite elemnt, from a single point of departure. Time-dependent element approximations are achieved with a progressive assembly of generalized elements method. Hyperbolic steady-state Euler algorithms and quasi-parabolic are spatial marching algorithms for solving the discretized equations. Application of the techniques is illustrated through calculation of the compressible flow around a hypersonic flight vehicle at 100,000 ft at zero angle of attack at speeds of Mach 4-10.

Spradley, L. W.↗

NASA NTRS1985

A high-precision finite element method for shock-tube calculations

A two-pass explicit scheme is developed in order to exploit some of the capabilities of finite difference modeling (FDM) for finite element modeling (FEM), which offers the opportunity to account for any type of geometry in fluid flow modeling. Features of the first-order upwind and the Lax-Wendroff high precision explicit finite difference algorithms are reviewed. A flux limiter is developed for FEM to serve as an analog for the single limiter function which has been defined for the various FDMs. It is shown that an antidiffusive limiter must be introduced into the weighting function which normally multiplies the time-derivative term in the variational equation. The two-pass scheme which results is demonstrated to be the equivalent of FDMs with five-point support. However, the present scheme is valid only for one-dimensional calculations and linear shape functions for shock tube flow phenomena. Further work is required for its use with nonlinear hyperbolic systems.

Hughes, T. J. R.↗

NASA NTRS1985

The LDEF interplanetary dust experiment

Explorer 46 data and the Long Duration Exposure Facility (LDEF) interplanetary dust experiment are examined. Analysis of the Explorer 46 data reveals the existence of particles of 0.1 micron and a mass of 1 x 10 to the -16th gm, the injection of the submicron particles directly by a comet (injection mass of about 5000 tons), and a submicron particle lifetime of about three years. The applications of LDEF data to particles in hyperbolic orbits, particle swarms, morning-to-evening asymmetry, the effects of the earth's orbit eccentricity, and the presence of interstellar dust are discussed. The effects of space debris on data collection are considered.

Singer, S. F.↗

NASA NTRS1985

Mass input into and output from the meteoritic complex

The consequences which arise from the mutual collisions occurring between interplanetary meteoroids, the Poynting-Robertson (PR) effect and the radiation pressure ejection of small meteoroids are examined. The size distribution and flux of micrometeoroids at 1 AU are derived and the dependence of spatial density on distance from the sun is established. The following conclusions are made: (1) the lifetimes of meteoroids with masses approximately greater than 0.00001 g are dominated by catastrophic collisions; (2) after bering crushed by collisions, 70 to 85 percent of this mass will be in the form of zodiacal light particles (with masses in the range of 10 to the -10th g to 10 to the -5th g) which will in part be transported by the PR effect towards the sun where they will evaporate; (3) the 15 to 30 percent of the collisional fragments which have masses approximately less than 10 to the -10th g will, for the most part, be injected into hyperbolic orbits by radiation pressure.

Gruen, E.↗

NASA NTRS1985

Information content in spectral calculations

Analytical procedures for extracting piecewise smooth solutions of hyperbolic systems from raw oscillatory data obtained by pseudospectral methods are developed. The validity of the approach is demonstrated for the case of linear problems with constant coefficients, and plausibility arguments are presented which indicate its applicability to nonlinear operators when the steady state has been achieved. Numerical results for the development of an oblique shock when a wedge is inserted at zero angle of attack into a uniform supersonic flow of an ideal gas (the time-dependent two-dimensional Euler equations discretized in space by the pseudospectral Chebyshev method) are presented in tables and graphs.

Abarbanel, S.↗

NASA NTRS1985

Recovering pointwise values of discontinuous data within spectral accuracy

The pointwise values of a function, f(x), can be accurately recovered either from its spectral or pseudospectral approximations, so that the accuracy solely depends on the local smoothness of f in the neighborhood of the point x. Most notably, given the equidistant function grid values, its intermediate point values are recovered within spectral accuracy, despite the possible presence of discontinuities scattered in the domain. (Recall that the usual spectral convergence rate decelerates otherwise to first order, throughout). To this end, a highly oscillatory smoothing kernel is employed in contrast to the more standard positive unit-mass mollifiers. In particular, post-processing of a stable Fourier method applied to hyperbolic equations with discontinuous data, recovers the exact solution modulo a spectrally small error. Numerical examples are presented.

Gottlieb, D.↗

NASA NTRS1986

Global collocation methods for approximation and the solution of partial differential equations

Polynomial interpolation methods are applied both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations. The derivative matrix for a general sequence of the collocation points is constructed. The approximate derivative is then found by a matrix times vector multiply. The effects of several factors on the performance of these methods including the effect of different collocation points are then explored. The resolution of the schemes for both smooth functions and functions with steep gradients or discontinuities in some derivative are also studied. The accuracy when the gradients occur both near the center of the region and in the vicinity of the boundary is investigated. The importance of the aliasing limit on the resolution of the approximation is investigated in detail. Also examined is the effect of boundary treatment on the stability and accuracy of the scheme.

Solomonoff, A.↗

NASA NTRS1986

On the Maxwellian distribution, symmetric form, and entropy conservation for the Euler equations

The Euler equations of gas dynamics have some very interesting properties in that the flux vector is a homogeneous function of the unknowns and the equations can be cast in symmetric hyperbolic form and satisfy the entropy conservation. The Euler equations are the moments of the Boltzmann equation of the kinetic theory of gases when the velocity distribution function is a Maxwellian. The present paper shows the relationship between the symmetrizability and the Maxwellian velocity distribution. The entropy conservation is in terms of the H-function, which is a slight modification of the H-function first introduced by Boltzmann in his famous H-theorem. In view of the H-theorem, it is suggested that the development of total H-diminishing (THD) numerical methods may be more profitable than the usual total variation diminishing (TVD) methods for obtaining wiggle-free solutions.

Deshpande, S. M.↗

NASA NTRS1986

A spline-based parameter estimation technique for static models of elastic structures

The problem of identifying the spatially varying coefficient of elasticity using an observed solution to the forward problem is considered. Under appropriate conditions this problem can be treated as a first order hyperbolic equation in the unknown coefficient. Some continuous dependence results are developed for this problem and a spline-based technique is proposed for approximating the unknown coefficient, based on these results. The convergence of the numerical scheme is established and error estimates obtained.

Dutt, P.↗

NASA NTRS1986

Survey of the status of finite element methods for partial differential equations

The finite element methods (FEM) have proved to be a powerful technique for the solution of boundary value problems associated with partial differential equations of either elliptic, parabolic, or hyperbolic type. They also have a good potential for utilization on parallel computers particularly in relation to the concept of domain decomposition. This report is intended as an introduction to the FEM for the nonspecialist. It contains a survey which is totally nonexhaustive, and it also contains as an illustration, a report on some new results concerning two specific applications, namely a free boundary fluid-structure interaction problem and the Euler equations for inviscid flows.

Temam, Roger↗

NASA NTRS1986

Comparison of mission design options for manned Mars missions

A number of manned Mars mission types, propulsion systems, and operational techniques are compared. Conjunction and opposition class missions for cryogenic, hybrid (cryo/storable), and NERVA propulsion concepts are addressed. In addition, both Earth and Mars orbit aerobraking, direct entry of landers, hyperbolic rendezvous, and electric propulsion cases are examined. A common payload to Mars was used for all cases. The basic figure of merit used was weight in low Earth orbit (LEO) at mission initiation. This is roughly proportional to launch costs.

Gus R Babb↗

NASA NTRS1986

Property of radially quadratic reflector systems

This report shows that when considered in terms of optical geometry, radially parabolic and radially hyperbolic mirrors used as mirrors for Cassegrain and parabolic antennas possess values similar to common conical horn reflector antennas.

Mizusawa, M.↗

NASA NTRS1987

Acoustic gravity waves: A computational approach

This paper discusses numerical solutions of a hyperbolic initial boundary value problem that arises from acoustic wave propagation in the atmosphere. Field equations are derived from the atmospheric fluid flow governed by the Euler equations. The resulting original problem is nonlinear. A first order linearized version of the problem is used for computational purposes. The main difficulty in the problem as with any open boundary problem is in obtaining stable boundary conditions. Approximate boundary conditions are derived and shown to be stable. Numerical results are presented to verify the effectiveness of these boundary conditions.

Hariharan, S. I.↗

NASA NTRS1987

Efficient implementation of essentially non-oscillatory shock capturing schemes

In the computation of discontinuous solutions of hyperbolic conservation laws, TVD (total-variation-diminishing), TVB (total-variation-bounded) and the recently developed ENO (essentially non-oscillatory) schemes have proven to be very useful. In this paper two improvements are discussed: a simple TVD Runge-Kutta type time discretization, and an ENO construction procedure based on fluxes rather than on cell averages. These improvements simplify the schemes considerably -- especially for multi-dimensional problems or problems with forcing terms. Preliminary numerical results are also given.

Shu, Chi-Wang↗