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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
NASA NTRS1980

Numerical methods for large-scale time-dependent partial differential equations

The factors that influence the choice of a numerical solution scheme for large-scale time-dependent partial differential equations are examined with emphasis on the treatment of boundaries. Consideration is given to standard boundary treatments, moving boundaries, and artificial boundaries simulating an infinite domain. Implicit methods and applications to specific problems are discussed with special attention to shocks and steady state solutions.

Turkel, E.↗
NASA NTRS1971

Numerical solution of stiff systems of ordinary differential equations with applications to electronic circuits

Systems of ordinary differential equations in which the magnitudes of the eigenvalues (or time constants) vary greatly are commonly called stiff. Such systems of equations arise in nuclear reactor kinetics, the flow of chemically reacting gas, dynamics, control theory, circuit analysis and other fields. The research reported develops an A-stable numerical integration technique for solving stiff systems of ordinary differential equations. The method, which is called the generalized trapezoidal rule, is a modification of the trapezoidal rule. However, the method is computationally more efficient than the trapezoidal rule when the solution of the almost-discontinuous segments is being calculated.

Rosenbaum, J. S.↗
NASA NTRS1982

Improved Numerical Differencing Analyzer

SINDA, Systems Improved Numerical Differencing Analyzer, solves differential and algebric equations representing physical systems. SINDA solves numerically almost any set of ordinary differential equations that represent transient behavior of a lumped-parameter system or any set of nonlinear algebraic equations that represents the steady state conditions of a physical system.

Skladany, J. T.↗
NASA NTRS1976

Conservation properties of numerical integration methods for systems of ordinary differential equations

If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.

Rosenbaum, J. S.↗
NASA NTRS1975

On the use of a finite difference method for solving anisotropic scattering problems

A new method of solving the radiative transfer equation is developed in which the scattering and absorption coefficients may have arbitrary variations with depth, and in which both internal (thermal) emission and incident radiation are allowed. Specular and diffuse reflection at both boundaries also is taken into account. The method begins by forming a paired set of coupled first-order differential equations for the symmetric and antisymmetric parts of the radiation field after writing the scattering integral as a numerical quadrature. These differential equations are broken into finite difference form, in which the symmetric and antisymmetric parts of the radiation field are found on alternate grid points. Numerical results for a number of test problems are shown, demonstrating that the method is very fast, that it returns specific intensities and fluxes that are accurate to at least a percent, and that it can be applied to optically thick problems.

Barkstrom, B. R.↗
NASA NTRS1972

Very high temperature laminar flow of a gas through the entrance region of a cooled tube - Numerical calculations and experimental results.

The laminar flow equations in differential form are solved numerically on a digital computer for flow of a very high temperature gas through the entrance region of an externally cooled tube. The solution method is described and calculations are carried out in conjunction with experimental measurements. The agreement with experiment is good, with the result indicating relatively large energy and momentum losses in the highly cooled flows considered where the pressure is nearly uniform along the flow and the core flow becomes non-adiabatic a few diameters downstream of the inlet. The effects of a large range of Reynolds number and Mach number (viscous dissipation) are also investigated.

Back, L. H.↗
NASA NTRS1997

Reflection Coefficients on Surfaces of Different Periodic Structure

Diffraction properties of lossy periodic gratings with the metal base were investigated by solving Maxwell's equations numerically using the differential method. Two periodic surfaces were employed in the simulation: triangle structure and tilting triangle structure. Based on the numerical solution and in conjunction with the algorithm of Adams-Moulton, we computed reflection coefficients of plane waves with different wavelengths and different incident angles. The dielectric properties were also explored using various dielectric constants. The results show that the reflection coefficients of both TE and TM waves are quite sensitive to the incident angles of the plane waves when the metal sheet exists, which is in good agreement with the experimental data.

Niu, Pengfei↗
NASA NTRS1985

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed.

Ortega, J. M.↗
NASA NTRS1985

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed.

Ortega, J. M.↗
NASA NTRS1995

More About Spurious Numerical Solutions Of DEs

Paper discusses reliability of time-dependent approach to numerical solution of nonlinear differential equations (DEs) that describe steady-state behaviors of physical systems. Time-dependent approach followed in related study described in "Spurious Numerical Solutions of Differential Equations" (ARC-13209).

Yee, H. C.↗
NASA NTRS1995

On the Importance of the Dynamics of Discretizations

It has been realized recently that the discrete maps resulting from numerical discretizations of differential equations can possess asymptotic dynamical behavior quite different from that of the original systems. This is the case not only for systems of Ordinary Differential Equations (ODEs) but in a more complicated manner for Partial Differential Equations (PDEs) used to model complex physics. The impact of the modified dynamics may be mild and even not observed for some numerical methods. For other classes of discretizations the impact may be pronounced, but not always obvious depending on the nonlinear model equations, the time steps, the grid spacings and the initial conditions. Non-convergence or convergence to periodic solutions might be easily recognizable but convergence to incorrect but plausible solutions may not be so obvious - even for discretized parameters within the linearized stability constraint. Based on our past four years of research, we will illustrate some of the pathology of the dynamics of discretizations, its possible impact and the usage of these schemes for model nonlinear ODEs, convection-diffusion equations and grid adaptations.

Sweby, Peter K.↗
NASA NTRS1990

Factorization and the synthesis of optimal feedback gains for distributed parameter systems

An approach based on Volterra factorization leads to a new methodology for the analysis and synthesis of the optimal feedback gain in the finite-time linear quadratic control problem for distributed parameter systems. The approach circumvents the need for solving and analyzing Riccati equations and provides a more transparent connection between the system dynamics and the optimal gain. The general results are further extended and specialized for the case where the underlying state is characterized by autonomous differential-delay dynamics. Numerical examples are given to illustrate the second-order convergence rate that is derived for an approximation scheme for the optimal feedback gain in the differential-delay problem.

Milman, Mark H.↗
NASA NTRS1991

Entropy-Based Approach To Nonlinear Stability

NASA technical memorandum suggests schemes for numerical solution of differential equations of flow made more accurate and robust by invoking second law of thermodynamics. Proposes instead of using artificial viscosity to suppress such unphysical solutions as spurious numerical oscillations and nonlinear instabilities, one should formulate equations so that rate of production of entropy within each cell of computational grid be nonnegative, as required by second law.

Merriam, Marshal L.↗
NASA NTRS1991

From differential to difference equations for first order ODEs

When constructing an algorithm for the numerical integration of a differential equation, one should first convert the known ordinary differential equation (ODE) into an ordinary difference equation. Given this difference equation, one can develop an appropriate numerical algorithm. This technical note describes the derivation of two such ordinary difference equations applicable to a first order ODE. The implicit ordinary difference equation has the same asymptotic expansion as the ODE itself, whereas the explicit ordinary difference equation has an asymptotic that is similar in structure but different in value when compared with that of the ODE.

Freed, Alan D.↗
NASA NTRS1973

A study of numerical methods of solution of the equations of motion of a controlled satellite under the influence of gravity gradient torque

Numerical methods of integration of the equations of motion of a controlled satellite under the influence of gravity-gradient torque are considered. The results of computer experimentation using a number of Runge-Kutta, multi-step, and extrapolation methods for the numerical integration of this differential system are presented, and particularly efficient methods are noted. A large bibliography of numerical methods for initial value problems for ordinary differential equations is presented, and a compilation of Runge-Kutta and multistep formulas is given. Less common numerical integration techniques from the literature are noted for further consideration.

Thompson, J. F.↗