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NASA NTRS1985

The experimental computer control of a two-dimensional hyperbolic system

The experimental computer control of a two-dimensional hyperbolic system is described. The system consists of a 5-foot gold-coated rubber membrane mounted on a circular cylindrical drum. Seven electrodes reside on a command surface located behind the membrane inside the drum. These electrodes served as capacitive sensors and electrostatic force actuators of transverse membrane deflection. The membrane was modelled as flat, isotropic and uniformly tensioned. Transverse membrane deflections were expanded in normal modes. Controllers regulating membrane deflection are designed using aggregation and design procedures based upon sensor and actuator influence functions. The resulting control laws are implemented on a minicomputer in two sets of experiments. The experimental study confirms the theoretically predicted behavior of the system, usefulness of the aggregation and design procedures, and the expectation that spillover can be made a beneficial source of damping in residual systems.

Yam, Y.↗

NASA NTRS1995

An approach to the development of numerical algorithms for first order linear hyperbolic systems in multiple space dimensions: The constant coefficient case

Two methods for developing high order single step explicit algorithms on symmetric stencils with data on only one time level are presented. Examples are given for the convection and linearized Euler equations with up to the eighth order accuracy in both space and time in one space dimension, and up to the sixth in two space dimensions. The method of characteristics is generalized to nondiagonalizable hyperbolic systems by using exact local polynominal solutions of the system, and the resulting exact propagator methods automatically incorporate the correct multidimensional wave propagation dynamics. Multivariate Taylor or Cauchy-Kowaleskaya expansions are also used to develop algorithms. Both of these methods can be applied to obtain algorithms of arbitrarily high order for hyperbolic systems in multiple space dimensions. Cross derivatives are included in the local approximations used to develop the algorithms in this paper in order to obtain high order accuracy, and improved isotropy and stability. Efficiency in meeting global error bounds is an important criterion for evaluating algorithms, and the higher order algorithms are shown to be up to several orders of magnitude more efficient even though they are more complex. Stable high order boundary conditions for the linearized Euler equations are developed in one space dimension, and demonstrated in two space dimensions.

Goodrich, John W.↗

DOE OSTI2018

Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. In conclusion, for time discretization, Crank–Nicolson scheme and backward Euler scheme are utilized.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities

We investigate a wide class of two-dimensional hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for the random process generated by sequences of dynamically Hölder observables. The observables could be unbounded, and the process may be non-stationary and need not have linearly growing variances. Our results apply to Anosov diffeomorphisms, Sinai dispersing billiards and their perturbations. The random processes under consideration are related to the fluctuation of Lyapunov exponents, the shrinking target problem, etc.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1981

Determining solutions of hyperbolic systems from incomplete data

An investigation is conducted regarding first-order hyperbolic systems of partial differential equations, taking into account problems for which complete initial data are not available. Problems of the considered kind arise in geophysical applications where satellites are used to collect data. In global weather prediction, it is possible to derive atmospheric temperature and pressure reasonably well over the whole globe from satellite measurements; obtaining the wind field globally is more difficult. It is pointed out that a simple model of atmospheric flow investigated in numerical weather prediction is governed by the shallow water equations. The effect of the Coriolis term on the linearized shallow-water equations is studied.

Bube, K. P.↗

NASA NTRS1980

The Split Coefficient Matrix method for hyperbolic systems of gasdynamic equations

The Split Coefficient Matrix (SCM) finite difference method for solving hyperbolic systems of equations is presented. This new method is based on the mathematical theory of characteristics. The development of the method from characteristic theory is presented. Boundary point calculation procedures consistent with the SCM method used at interior points are explained. The split coefficient matrices that define the method for steady supersonic and unsteady inviscid flows are given for several examples. The SCM method is used to compute several flow fields to demonstrate its accuracy and versatility. The similarities and differences between the SCM method and the lambda-scheme are discussed.

Chakravarthy, S. R.↗

NASA NTRS1983

Computational methods for estimation of parameters in hyperbolic systems

Approximation techniques for estimating spatially varying coefficients and unknown boundary parameters in second order hyperbolic systems are discussed. Methods for state approximation (cubic splines, tau-Legendre) and approximation of function space parameters (interpolatory splines) are outlined and numerical findings for use of the resulting schemes in model "one dimensional seismic inversion' problems are summarized.

Banks, H. T.↗

NASA NTRS1986

Estimation of discontinuous coefficients and boundary parameters for hyperbolic systems

The problem of estimating discontinuous coefficients, including locations of discontinuities, that occur in second order hyperbolic systems typical of those arising in I-D surface seismic problems is discussed. In addition, the problem of identifying unknown parameters that appear in boundary conditions for the system is treated. A spline-based approximation theory is presented, together with related convergence findings and representative numerical examples.

Lamm, P. K.↗

NASA NTRS1982

Conservative supra-characteristics method for splitting the hyperbolic systems of gasdynamics for real and perfect gases

A conservative flux difference splitting is presented for the hyperbolic systems of gasdynamics. The stable robust method is suitable for wide application in a variety of schemes, explicit or implicit, iterative or direct, for marching in either time or space. The splitting is modeled on the local quasi one dimensional characteristics system for multi-dimensional flow similar to Chakravarthy's nonconservative split coefficient matrix method; but, as the result of maintaining global conservation, the method is able to capture sharp shocks correctly. The embedded characteristics formulation is cast in a primitive variable the volumetric internal energy (rather than the pressure) that is effective for treating real as well as perfect gases. Finally the relationship of the splitting to characteristics boundary conditions is discussed and the associated conservative matrix formulation for a computed blown wall boundary condition is developed as an example. The theoretical development employs and extends the notion of Roe of constructing stable upwind difference formulae by sending split simple one sided flux difference pieces to appropriate mesh sites. The developments are also believed to have the potential for aiding in the analysis of both existing and new conservative difference schemes.

Lombard, C. K.↗

NASA NTRS1988

Estimation of discontinuous coefficients and boundary parameters for hyperbolic systems

The problem of estimating discontinuous coefficients, including locations of discontinuities, that occur in second-order hyperbolic systems typical of those arising in one-dimensional surface seismic problems, is considered. The problem of identifying unknown parameters that appear in boundary conditions for the system is addressed. A spline-based approximation theory is presented as well as related convergence findings and representative numerical examples.

Lamm, Patricia K.↗

NASA NTRS1990

Time-dependent boundary conditions for hyperbolic systems. II

A general boundary condition formalism is developed for all types of boundary conditions to which hyperbolic systems are subject; the formalism makes possible a 'cookbook' approach to boundary conditions, by means of which novel boundary 'recipes' may be derived and previously devised ones may be consulted as required. Numerous useful conditions are derived for such CFD problems as subsonic and supersonic inflows and outflows, nonreflecting boundaries, force-free boundaries, constant pressure boundaries, and constant mass flux. Attention is given to the computation and integration of time derivatives.

Thompson, Kevin W.↗

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.↗

DOE OSTI2017

On the arithmetic intensity of high-order finite-volume discretizations for hyperbolic systems of conservation laws

It has been conjectured that higher-order discretizations for partial differential equations will have advantages over the lower-order counterparts commonly used today. The reasoning is that the increase in arithmetic operations will be more than offset by the reduction in data transfers and the increase in concurrent floating-point units. To evaluate this conjecture, the arithmetic intensity of a class of high-order finite-volume discretizations for hyperbolic systems of conservation laws is theoretically analyzed for spatial discretizations from orders three through eight in arbitrary dimensions. Additionally, three cache models are considered: the limiting cases of no cache and infinite cache as well as a finite-sized cache model. Models are validated experimentally by measuring floating-point operations and data transfers on an IBM Blue Gene/Q node. Theory and experiments demonstrate that high-order finite-volume methods will be able to provide increases in arithmetic intensity that will be necessary to make better utilization of on-node floating-point capability.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1976

An implicit finite-difference algorithm for hyperbolic systems in conservation-law form

An implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation-law form. The algorithm is second-order time-accurate, noniterative, and in a spatially factored form. Second- or fourth-order central and second-order one-sided spatial differencing are accommodated within the solution of a block tridiagonal system of equations. Significant conceptual and computational simplifications are made for systems whose flux vectors are homogeneous functions (of degree one), e.g., the Eulerian gasdynamic equations. Conservative hybrid schemes, which switch from central to one-sided spatial differencing whenever the local characteristic speeds are of the same sign, are constructed to improve the resolution of weak solutions. Numerical solutions are presented for a nonlinear scalar model equation and the two-dimensional Eulerian gasdynamic equations.

Beam, R. M.↗

NASA NTRS1984

Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations

A Petrov-Galerkin finite element formulation is presented for first-order hyperbolic systems of conservation laws with particular emphasis on the compressible Euler equations. Applications of the methodology are made to one- and two-dimensional steady and unsteady flows with shocks. Results obtained suggest the potential of the type of methods developed.

Hughes, T. J. R.↗

DOE OSTI2018

Cone-grid scheme for solving hyperbolic systems of conservation laws and one application

In this work, we introduce a new numerical method, the so-called cone-grid scheme for solving hyperbolic systems of conservation laws. The formulation and the theoretical analysis for this scheme are slightly simpler than other algorithms. Simultaneously, our technique leads to a more efficacious numerical scheme. The Riemann solution is the basic ingredient for this scheme. We also give the main application of this technique, namely the global weak solutions for the one-dimensional model prescribes heat conduction in solids at low temperature, which called phonon–Bose model. This system consists of a conservation equation for the energy density and the heat flux. The scheme also satisfies positivity. The cone-grid scheme was compared with exact solution by three numerical examples, where explicit solutions are known. These numerical results verify the accuracy of the proposed scheme qualitatively and quantitatively.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1986

Estimation of coefficients and boundary parameters in hyperbolic systems

Semi-discrete Galerkin approximation schemes in connection with inverse problems for the estimation of spatially varying coefficients and boundary condition parameters in second order hyperbolic systems typical of those arising in 1-D surface seismic problems are considered. Spline based algorithms are proposed for which theoretical convergence results along with a representative sample of numerical findings are given.

Banks, H. T.↗

DOE OSTI2024

Uniform accuracy of implicit-explicit Runge-Kutta (IMEX-RK) schemes for hyperbolic systems with relaxation

Implicit-explicit Runge-Kutta (IMEX-RK) schemes are popular methods to treat multiscale equations that contain a stiff part and a non-stiff part, where the stiff part is characterized by a small parameter. Here, in this work, we prove rigorously the uniform stability and uniform accuracy of a class of IMEX-RK schemes for a linear hyperbolic system with stiff relaxation. The result we obtain is optimal in the sense that it holds regardless of the value of and the order of accuracy is the same as the design order of the original scheme, i.e., there is no order reduction.

97 MATHEMATICS AND COMPUTING↗