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All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer

Recently, Berenger introduced a Perfectly Matched Layer (PML) technique for absorbing electromagnetic waves. In the present paper, a perfectly matched layer is proposed for absorbing out-going two-dimensional waves in a uniform mean flow, generated by linearized Euler equations. It is well known that the linearized Euler equations support acoustic waves, which travel with the speed of sound relative to the mean flow, and vorticity and entropy waves, which travel with the mean flow. The PML equations to be used at a region adjacent to the artificial boundary for absorbing these linear waves are defined. Plane waves solutions to the PML equations are developed and wave propagation and absorption properties are given. It is shown that the theoretical reflection coefficients at an interface between the Euler and PML domains are zero, independent of the angle of incidence and frequency of the waves. As such, the present study points out a possible alternative approach for absorbing out-going waves of the Euler equations with little or no reflection in computation. Numerical examples that demonstrate the validity of the proposed PML equations are also presented.

Hu, Fang Q.↗
NASA NTRS1985

Variability simulations with a steady, linearized primitive equations model

Solutions of the steady, primitive equations on a sphere, linearized about a zonally symmetric basic state are computed for the purpose of simulating monthly mean variability in the troposphere. The basic states are observed, winter monthly mean, zonal means of zontal and meridional velocities, temperatures and surface pressures computed from the 15 year NMC time series. A least squares fit to a series of Legendre polynomials is used to compute the basic states between 20 H and the equator, and the hemispheres are assumed symmetric. The model is spectral in the zonal direction, and centered differences are employed in the meridional and vertical directions. Since the model is steady and linear, the solution is obtained by inversion of a block, pente-diagonal matrix. The model simulates the climatology of the GFDL nine level, spectral general circulation model quite closely, particularly in middle latitudes above the boundary layer. This experiment is an extension of that simulation to examine variability of the steady, linear solution.

Kinter, J. L., III↗
NASA NTRS1988

Modified non-linear Burgers' equations and cosmic ray shocks

A reductive perturbation scheme is used to derive a generalized non-linear Burgers' equation, which includes the effects of dispersion, in the long wavelength regime for the two-fluid hydrodynamical model used to describe cosmic ray acceleration by the first-order Fermi process in astrophysical shocks. The generalized Burger's equation is derived for both relativistic and non-relativistic cosmic ray shocks, and describes the time evolution of weak shocks in the theory of diffusive shock acceleration. The inclusion of dispersive effects modifies the phase velocity of the shock obtained from the lower order non-linear Burger's equation through the introduction of higher order terms from the long wavelength dispersion equation. The travelling wave solution of the generalized Burgers' equation for a single shock shows that larger cosmic ray pressures result in broader shock transitions. The results for relativistic shocks show a steepening of the shock as the shock speed approaches the relativistic cosmic ray sound speed. The dependence of the shock speed on the cosmic ray pressure is also discussed.

Zank, G. P.↗
NASA NTRS1973

Sparse matrix methods based on orthogonality and conjugacy

A matrix having a high percentage of zero elements is called spares. In the solution of systems of linear equations or linear least squares problems involving large sparse matrices, significant saving of computer cost can be achieved by taking advantage of the sparsity. The conjugate gradient algorithm and a set of related algorithms are described.

Lawson, C. L.↗
NASA NTRS1982

An analytic approach to two-fixed-impulse transfers between Keplerian orbits

Solutions are obtained for the two-impulse transfer of a vehicle between arbitrary inclined orbits in an inverse-square force field with the restriction that the magnitude of each of the two impulses has a fixed preassigned value. The two magnitudes need not be equal. The equations for the conservation of angular momentum and energy are augmented by the Laplace integral; these equations establish linear relationships between several of the variables. This set of linear equations and one of two quadratic equations constitute the analytically tractable part of the solution. The remaining part, consisting of finding the zeros (if they exist) of a single trigonometric function of one variable, is solved using numerical methods. Explicit lower bounds on each of the magnitudes of the two impulses are obtained by requiring the solution be real. Graphical results are presented to illustrate the solution.

Jezewski, D. J.↗
NASA NTRS1993

Some aspects of high-order numerical solutions of the linear convection equation with forced boundary conditions

A six-stage low-storage Runge-Kutta time-marching method is presented and shown to be an efficient method for use with high-accuracy spatial difference operators for wave propagation problems. The accuracy of the method for inhomogeneous ordinary differential equations is demonstrated through numerical solutions of the linear convection equation with forced boundary conditions. Numerical experiments are presented simulating a sine wave and a Gaussian pulse propagating into and through the domain. For practical levels of mesh refinement corresponding to roughly ten points per wavelength, the six-stage Runge-Kutta method is more accurate than the popular fourth-order Runge-Kutta method. Further numerical experiments are presented which show that the numerical boundary scheme at an inflow boundary can be a significant source of error when high-accuracy spatial discretizations are used.

Zingg, D. W.↗
NASA NTRS1977

Exact solution of some linear matrix equations using algebraic methods

A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.

Djaferis, T. E.↗
DOE OSTI2023

Quantum algorithm for the linear Vlasov equation with collisions

The Vlasov equation is a nonlinear partial differential equation that provides a first-principles description of the dynamics of plasmas. Its linear limit is routinely used in plasma physics to investigate plasma oscillations and stability. In this paper, we present a quantum algorithm that simulates the linearized Vlasov equation with and without collisions, in the one-dimensional electrostatic limit. Rather than solving this equation in its native spatial and velocity phase space, we adopt an efficient representation in the dual space yielded by a Fourier-Hermite expansion. For a given simulation time, the Fourier-Hermite representation is exponentially more compact, thus yielding a classical algorithm that can match the performance of a previously proposed quantum algorithm for this problem. Further, this representation results in a system of linear ordinary differential equations (ODEs) which can be solved with well-developed quantum algorithms: a Hamiltonian simulation in the collisionless case, and quantum ODE solvers in the collisional case. In particular, we demonstrate that a quadratic speedup in system size is attainable.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1991

The linear Boltzmann equation in slab geometry - Development and verification of a reliable and efficient solution

The linear Boltzmann equation can be cast in a form mathematically identical to the radiation-transport equation. A multigroup procedure is used to reduce the energy (or velocity) dependence of the transport equation to a series of one-speed problems. Each of these one-speed problems is equivalent to the monochromatic radiative-transfer problem, and existing software is used to solve this problem in slab geometry. The numerical code conserves particles in elastic collisions. Generic examples are provided to illustrate the applicability of this approach. Although this formalism can, in principle, be applied to a variety of test particle or linearized gas dynamics problems, it is particularly well-suited to study the thermalization of suprathermal particles interacting with a background medium when the thermal motion of the background cannot be ignored. Extensions of the formalism to include external forces and spherical geometry are also feasible.

Stamnes, K.↗
NASA NTRS1979

Exact solution of some linear matrix equations using algebraic methods

Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.

Djaferis, T. E.↗
NASA NTRS1989

Nonlinear strain-displacement relations and flexible multibody dynamics

Dynamics of chains of flexible bodies undergoing large rigid body motions, but small elastic deflections are considered. The role of nonlinear strain-displacement relations in the development of the motion equations correct to first order in elastic deflections is investigated. The general form of these equations linearized only in the small elastic deflections is presented, and the relative significance of various nonlinear terms is studied both analytically and through the use of the numerical simulations. Numerical simulations are performed for a two link chain constrained to move in the plane, subject to hinge torques. Each link is modeled as a thin beam. Slew maneuver simulation results are compared for models with and without properly modeled kinematics of deformation. The goal of this case study is to quantify the importance of the terms in the equations of motion which arise from the inclusion of nonlinear strain-displacement relations. It is concluded that unless the consistently linearized equations in elastic deflections and speeds are available and necessary, the inconsistently (prematurely) linearized equations should be replaced in all cases by ruthlessly linearized equations: equations in which all nonlinear terms involving the elastic deflections and speeds are ignored.

Padilla, Carlos E.↗
NASA NTRS2002

Multidimensional indexing structure for use with linear optimization queries

Linear optimization queries, which usually arise in various decision support and resource planning applications, are queries that retrieve top N data records (where N is an integer greater than zero) which satisfy a specific optimization criterion. The optimization criterion is to either maximize or minimize a linear equation. The coefficients of the linear equation are given at query time. Methods and apparatus are disclosed for constructing, maintaining and utilizing a multidimensional indexing structure of database records to improve the execution speed of linear optimization queries. Database records with numerical attributes are organized into a number of layers and each layer represents a geometric structure called convex hull. Such linear optimization queries are processed by searching from the outer-most layer of this multi-layer indexing structure inwards. At least one record per layer will satisfy the query criterion and the number of layers needed to be searched depends on the spatial distribution of records, the query-issued linear coefficients, and N, the number of records to be returned. When N is small compared to the total size of the database, answering the query typically requires searching only a small fraction of all relevant records, resulting in a tremendous speedup as compared to linearly scanning the entire dataset.

Bergman, Lawrence David↗
NASA NTRS1966

Numerical Studies of the Non-linear Vlasov Equation

The subject of this dissertation is the numerical integration of the initial-value problem for the non-linear Vlasov equation. The Vlasov equation is used to describe the dynamics of a "collisionless", one-dimensional, classical electron gas confined between two perfectly reflecting boundaries. Only the long-range Coulomb interactions of the electrons are considered; effects associated with the discrete structure are neglected. The numerical results obtained for non-linear Landau damping compare well with similar results obtained by Knorr. A general statement of the results on stable initial conditions is: As the degree of non-linearity of the initial conditions is increased, the deviation from linear Landau damping appears sooner and is more severe. In some cases damping was observed to cease. Curves showing the time dependence of the damping decrement are derived and compared with predictions of non-linear theories. New results obtained in this study include the observation that for strongly non-linear cases, the damping of the electric field causes an initially Maxwellian fo (v, O) to develop a peak in the neighborhood of the phase velocity; strong growth of the second harmonic is seen after fo (v, t) develops such a peak. Also new in this study is the interpretation of the development of a certain class of strongly unstable initial conditions as approaching an inhomogeneous equilibrium.

NUMERICAL INTEGRATION↗
NASA NTRS1972

Liapunov functions for non-linear difference equation stability analysis.

Liapunov functions to determine the stability of non-linear autonomous difference equations can be developed through the use of auxiliary exact difference equations. For this purpose definitions are introduced for the gradient of an implicit function of a discrete variable, a principal sum, a definite sum and an exact difference equation, and a theorem for exactness of a difference form is proved. Examples illustrate the procedure.

Park, K. E.↗