Results for “HYPERBOLIC EQUATION” — Page 4

NASA NTRS1972

Computation of space shuttle flow fields using noncentered finite-difference schemes.

Second- and third-order, noncentered finite-difference schemes are described for the numerical solution of the hyperbolic equations of fluid dynamics. The advantages of noncentered methods over the more conventional centered schemes are: simpler programming logic, nonhomogeneous terms are easily included, and generalization to multidimensional problems is direct. Second- and third-order methods are compared with regard to dissipative and dispersive errors and shock-capturing ability. These schemes are then used in a shock-capturing technique to determine the inviscid, supersonic flow field surrounding space shuttle vehicles (SSV). Resulting flow fields about typical pointed and blunted, delta-winged SSVs at angle of attack are presented and compared with experiment.

Kutler, P.↗

NASA NTRS1977

On the stability of Galerkin methods for initial-boundary value problems for hyperbolic systems

The stability of approximating the solution of mixed initial-boundary value problems for hyperbolic systems by semidiscrete Galerkin methods is studied. It is shown that a particular straightforward Galerkin method yields an unstable approximation, and that this numerical instability is caused by an improper treatment of the boundary. Stable schemes are then presented, one of which differs from the unstable scheme only insofar as the treatment of the boundary is concerned. These stable schemes make use of a particular matrix which symmetrizes the differential system. It is therefore shown that the use of this matrix is crucial to the stability of the computations as well as for obtaining a priori bounds on the energy of the continuous system. This symmetrizing matrix is also related to the diagonalizing matrix for the system of hyperbolic equations and to the Liapunov matrix for the system of ordinary differential equations resulting from the application of Galerkin's method.

Gunzburger, M. D.↗

NASA NTRS1978

Finite elements and finite differences for transonic flow calculations

The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.

Hafez, M. M.↗

NASA NTRS1979

Error analysis of finite difference schemes applied to hyperbolic initial boundary value problems

Finite difference methods for the numerical solution of mixed initial boundary value problems for hyperbolic equations are studied. The reported investigation has the objective to develop a technique for the total error analysis of a finite difference scheme, taking into account initial approximations, boundary conditions, and interior approximation. Attention is given to the Cauchy problem and the initial approximation, the homogeneous problem in an infinite strip with inhomogeneous boundary data, the reflection of errors in the boundaries, and two different boundary approximations for the leapfrog scheme with a fourth order accurate difference operator in space.

Skollermo, G.↗

NASA NTRS1981

The stability of pseudospectral-Chebyshev methods

The pseudospectral-Chebyshev methods are shown to be convergent in variable coefficient problems and, in some cases, hyperbolic problems. The analysis demonstrates that the rate of convergence is greater for finite difference methods or the finite element method. For a single first-order hyperbolic equation, the method is seen as remaining stable even when the coefficient changes sign, although in this case it is specified that care must be taken to have adequate spatial resolution. It is noted that this fact, combined with the fact that collocation methods are easy to apply in the nonlinear case, shows that the pseudospectral method is in general preferable to the Galerkin or Tau methods.

Gottlieb, D.↗

NASA NTRS1983

Computations of two-phase supersonic nozzle flows by a space-marching method

A practical method for efficiently analyzing the two-phase flow field in the supersonic region of an axisymmetric rocket nozzle is presented. The governing hyperbolic equations for the gas- and particle-phases are simultaneously integrated downstream using a space-marching procedure. Both one- and two-phase flow computations are performed for the JPL axisymmetric nozzle. The results show the nonequilibrium effects which are due to the existence of particles in the flow. The comparisons with the experimental data and other numerical solutions are made to examine the accuracy of the present method.

Fujii, K.↗

NASA NTRS1984

Relaxation methods for unfactored implicit upwind schemes

Relaxation methods are presented for unfactored implicit upwind schemes for hyperbolic equations. The theoretical bases are explained using linear and nonlinear scalar equations; construction of the method for the unsteady Euler equations (nonlinear system) is but a natural extension. One of the important advantages of the above methods vis a vis factored implicit schemes is the possibility of faster convergence to steady state, as illustrated by the results. Several classes of relaxation schemes such as pointwise, linewise, Gauss-Seidel, and non-Gauss-Seidel methods are discussed, along with various strategies for convergence.

Chakravarthy, S. R.↗

NASA NTRS1985

The eigenvalues of the pseudospectral Fourier approximation to the operator sin (2x) d/dx

It is shown that the eigenvalues Z sub i of the pseudospectral Fourier approximation to the operator sin(2x) curly d/curly dx satisfy (R sub e) (Z sub i) = + or - 1 or (R sub e)(Z sub I) = 0. Whereas this does not prove stability for the Fourier method, applied to the hyperbolic equation U sub t = sin (2x)(U sub x) - pi x pi; it indicates that the growth in time of the numerical solution is essentially the same as that of the solution to the differential equation.

Tal-Ezer, H.↗

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 NTRS1985

Large-scale computations in fluid mechanics; Proceedings of the Fifteenth Summer Seminar on Applied Mathematics, University of California, La Jolla, CA, June 27-July 8, 1983. Parts 1 & 2

Papers are presented on such topics as the use of semi-Lagrangian advective schemes in meteorological modeling; computation with high-resolution upwind schemes for hyperbolic equations; dynamics of flame propagation in a turbulent field; a modified finite element method for solving the incompressible Navier-Stokes equations; computational fusion magnetohydrodynamics; and a nonoscillatory shock capturing scheme using flux-limited dissipation. Consideration is also given to the use of spectral techniques in numerical weather prediction; numerical methods for the incorporation of mountains in atmospheric models; techniques for the numerical simulation of large-scale eddies in geophysical fluid dynamics; high-resolution TVD schemes using flux limiters; upwind-difference methods for aerodynamic problems governed by the Euler equations; and an MHD model of the earth's magnetosphere.

Engquist, B. E.↗

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

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 NTRS1987

Discontinuous solutions to hyperbolic systems under operator splitting

Two-dimensional systems of linear hyperbolic equations are studied with regard to their behavior under a solution strategy that in alternate time-steps solves exactly the component one-dimensional operators. The initial data is a step function across an oblique discontinuity. The manner in which this discontinuity breaks up under repeated applications of the split operator is analyzed, and it is shown that the split solution will fail to match the true solution in any case where the two operators do not share all their eigenvectors. The special case of the fluid flow equations is analyzed in more detail, and it is shown that arbitrary initial data gives rise to pseudo acoustic waves and a non-physical stationary wave. The implications of these findings for the design of high-resolution computing schemes are discussed.

Roe, P. L.↗

NASA NTRS1987

Spectral collocation methods

This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2.

Hussaini, M. Y.↗

NASA NTRS1987

Time dependent boundary conditions for hyperbolic systems

Nonreflecting boundary conditions are defined for multidimensional fluid dynamics problems where waves enter and leave the interior of a domain modeled by hyperbolic equations. Separate equations are defined for each type of incoming and outgoing wave. Temporally varying problems are considered in terms of a nonreflecting boundary condition which permit the amplitude of incoming waves to remain constant over time. Conservative expressions are presented that include dissipative terms. Applications of the computational techniques are illustrated with sample results for a traveling shock wave, a shock tube, a spherical explosion and expansion problems on one- and two-dimensions.

Thompson, Kevin W.↗

NASA NTRS1988

An adaptive pseudospectral method for discontinuous problems

The accuracy of adaptively chosen, mapped polynomial approximations is studied for functions with steep gradients or discontinuities. It is shown that, for steep gradient functions, one can obtain spectral accuracy in the original coordinate system by using polynomial approximations in a transformed coordinate system with substantially fewer collocation points than are necessary using polynomial expansion directly in the original, physical, coordinate system. It is also shown that one can avoid the usual Gibbs oscillation associated with steep gradient solutions of hyperbolic pde's by approximation in suitably chosen coordinate systems. Continuous, high gradient solutions are computed with spectral accuracy (as measured in the physical coordinate system). Discontinuous solutions associated with nonlinear hyperbolic equations can be accurately computed by using an artificial viscosity chosen to smooth out the solution in the mapped, computational domain. Thus, shocks can be effectively resolved on a scale that is subgrid to the resolution available with collocation only in the physical domain. Examples with Fourier and Chebyshev collocation are given.

Augenbaum, Jeffrey M.↗

NASA NTRS1989

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 NTRS1989

Spectral collocation methods

This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2.

Hussaini, M. Y.↗